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IN   MEMORIAM 
FLORIAN  CAJORI 


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ECLECTIC  EDUCATIONAL  SLlIES. 


(fcUmnttaru  |.Igtka. 


liAY^S   ALGEBRA, 

PART    FIRST: 

ON    THE 

ANALYTIC  AND  INDUCTIVE 

METHODS    OF    INSTRUCTION: 

WITH 

NUMEROUS    PRACTICAL    EXERCISES 

DESIGNED     FOn 

COMMON  SCHOOLS  AND  ACADEMIES. 


BY  JOSEPH  RAY,  M.  D. 

PROFLSSOR  OF   MATHPIMATICS  IN  WOODWARD  COLLEO£. 


REVISED    EDITION. 


VAN  ANTWERP,  BRAGG  &  CO., 

137   WALNUT  STREET,  28  BOND  STREET, 

CINCINNATT.  NEW  YORK. 


ECLECTIC  EDUCATIONAL  SERIES. 


RAY'S  MATHEMATICS. 

EMBRACING 

A  Thorough,  Pfogressive,  and  Complete  Course  in  Arith^netic,  Algebra^ 
and  the  Higher  Mafhemalics. 

Ray's  Primary  Arithmetic.  Ray's  Higher  Arithmetic. 

Ray's  Intellectual  Arithmetic.  Key  to  Ray's  Higher. 

Ray's  Practical  Arithmetic.  Ray's  New  Elementary  Algebra. 

Key  to  Ray's  Arithmetics.  Ray's  New  Higher  Algebra. 

Ray's  Test  Examples  in  Arith.  Key  to  Ray's  New  Algebras. 


Raifs  Plane  and  Solid  Geometry. 

By  Eli  T.  Tappan,  A.  M.,  Fres't  Kenyon  College.  l2mo.,  clotli, 
276  pp. 

Raifs  Geometry  and  Trigonometry. 

By  Eli  T.  Tappan,  A.  M. ,  Preset  Kenyon  College.  8vo. ,  sheep,  420  pp. 

Ray's  Analytic  Geometry. 

By  Geo.  H.  Howison,  A.  M.,Prof.  in  Mass.  Institute  of  Technology. 
TreatLse  on  Analytic  Geometry,  especially  as  applied  to  the  prop- 
erties of  Conies:  including  the  Modern  Methods  of  Abridged 
Notation.     8vo.,  sheep,  574  pp. 

Ray's  Elements  of  Astronomy. 

By  S.  H.  Peabody,  A.  M.,  Prof,  in  the  Chicago  High  School, 
Handsomely  and   profusely  illustrated.     8vo.,  sheep,  336  pp. 

Ray's  Surveying  and  Navigation. 

With  a  Preliminary  Treatise  on  Trigonometry  and  Mensuration. 
By  A.  Schuyler,  Prof,  of  Applied  Mathematics  and  Logic  in  Bald- 
win University.     Svo.,  sheep,  403  pp. 

Ray's  differential  and  Integral  Caleulus. 

Elements  of  the  Infinitesimal  Calculus,  with  numerous  examples 
and  api)lications  to  Analysis  and  Geometry,  By  J  as.  G.  Clark,  A. 
M.,  /^jv/.  in  William  Jewell  College.     8vo.,  slieep,  440  pj). 


Entered  according  taAct  of  Congress,  in  the  ye.ir  1848,  by  Winthrop  B.  Smith, 

in  the  Clerk's  Office  of  the  District  Court  of  the  United 

States  for  the  District  of  Ohio. 


K2.L 


PREFACE. 


The  object  of  the  study  of  Mathematics,  is  two  fold— the  acqui- 
sition of  useful  knowledge,  and  the  cultivation  and  discipline  of 
the  mental  powers.  A  parent  often  inquires,  "Why  should  my 
son  study  mathematics?  I  do  not  expect  him  to  be  a  surveyor,  an 
engineer,  or  an  astronomer."  Yet,  the  parent  is  very  desirous 
that  his  son  should  be  able  to  reason  correctly,  and  to  exercise, 
in  all  his  relations  in  life,  the  energies  of  a  cultivated  and  disci- 
plined mind.  This  is,  indeed,  of  more  value  than  the  mere  attain- 
ment of  any  branch  of  knowledge. 

The  science  of  Algebra,  properly  taught,  stands  among  the  first 
of  those  studies  essential  to  both  the  great  objects  of  education. 
In  a  course  of  instruction  properly  arranged,  it  naturally  follows 
Arithmetic,  and  should  be  taught  immediately  after  it. 

In  the  following  work,  the  object  has  been,  to  furnish  an  ele- 
mentary treatise,  commencing  with  the  first  principles,  and  leading 
the  pupil,  by  gradual  and  easy  steps,  to  a  knowledge  of  the  ele- 
ments of  the  science.  The  design  has  been,  to  present  these  in  a 
brief,  clear,  and  scientific  manner,  so  that  the  pupil  should  not  be 
taught  merely  to  perform  a  certain  routine  of  exercises  mechani- 
cally, but  to  understand  the  wJit/  and  the  wherefore  of  every  step. 
For  this  purpose,  every  rule  is  demonstrated,  and  every  principle 
analyzed,  in  order  that  the  mind  of  the  pupil  may  be  disciplined 
and  strengthened  so  as  to  prepare  him,  either  for  pursuing  the 
study  of  Mathematics  intelligently,  or  more  successfully  attending 
to  any  pursuit  in  life. 

Some  teachers  may  object,  that  this  work  is  too  simple,  and  too 
easily  understood.  A  leading  object  has  been,  to  make  the  pupil 
feel,  that  he  is  not  operating  on  unmeaning  symbols,  by  means  of 
arbitrary  rules ;  that  Algebra  is  both  a  rational  and  a  practical 
subject,  and  that  he  can  rely  upon  his  reasoning,  and  the  results 

3 


^♦^►ri«  ikWO^-y 


IV  PREFACE. 


of  his  operations,  with  the  same  confidence  as  in  arithmetic.  For 
this  purpose,  he  is  furnished,  at  almost  every  step,  with  the  means 
of  testing  the  accuracy  of  the  principles  on  which  the  rules  are 
founded,  and  of  the  results  which  they  produce. 

Throughout  the  Avork,  the  aim  has  been,  to  combine  the  clear, 
explanatory  methods  of  the  French  mathematicians,  with  the  prac- 
tical exercises  of  the  English  and  German,  so  that  the  pupil  should 
acquire  both  a  practical  and  theoretical  knowledge  of  the  subject. 

While  every  page  is  the  result  of  the  author's  own  reflection, 
and  the  experience  of  many  years  in  the  school-room,  it  is  also 
proper  to  state,  that  a  large  number  of  the  best  treatises  on  the 
same  subject,  both  English  and  French,  have  been  carefully  con- 
sulted, so  that  the  present  work  might  embrace  the  modern  and 
most  approved  methods  of  treating  the  various  subjects  presented. 

With  these  remarks,  the  work  is  sul)mitted  to  the  judgment  of 
fellow  laborers  in  the  field  of  education. 

Woodward  College,  August,  1848. 


SUGGESTIONS     TO    TEACHERS. 

It  is  intended  that  the  pupil  shall  recite  the  Intellectual  Exercises  with 
the  book  open  before  him,  as  in  mental  Arithmetic.  Advanced  pupils  may 
omit  these  exercises. 

The  following  subjects  may  be  omitted  by  the  younger  pupils,  and  passed 
over  by  those  more  advanced,  until  the  book  is  revicAved. 

Observations  on  Addition  and  Subtraction,  Articles  60 — 64. 

The  greater  part  of  Chapter  11. 

Supplement  to  Equations  of  the  First  Degree,  Articles  164 — 177. 

Properties  of  the  Roots  of  an  Equation  of  the  Second  Degree,  Articles 
215—217. 

In  reviewing  the  book,  the  pupil  should  demonstrate  the  rules  on  the 
blackboard. 

The  work  will  bo  found  to  contain  a  large  number  of  examples  for  prac- 
tice. Should  any  instructor  deem  these  too  nixmerous,  a  portion  of  them 
may  bo  omitted. 

To  teach  the  subject  successfully,  the  principles  must  be  first  clearly 
explained,  and  then  the  pupil  exercised  in  the  solution  of  appropriate 
examples,  until  they  are  rendered  perfectly  familiar. 


CONTENTS. 


ARTICLES. 

Intellectual  Exercises,  XIV  Lessons, 

CHAPTER  I— FUNDAMENTAL  RULES. 

Preliminary  Definitions  and  Principles 1 —  15 

Definitions  of  Terms,  and  Explanation  of  Siijns 16—  52 

Examples  to  illustrate  the  use  of  the  Signs 

Addition 53 —  55 

Subtraction 56 —  59 

Observations  on  Addition  and  Subtraction 60 —  64 

Multiplication— Rule  of  the  Coefficients 65 —  67 

Rule  of  the  Exponents 69 

General  Rule  for  the  Signs 72 

General  Rule  for  Multiplication 

Division  of  Monomials— Rule  of  the  Signs 73 —  75 

Polynomials — Rule 79 

CHAPTER  II— THEOREMS,  FACTORING,  &c. 

Algebraic  Theorems 80—  86 

Factoring 87—  96 

Greatest  Common  Divisor 97 — 106 

Least  Common  Multiple 107—112 

CHAPTER  III— ALGEBRAIC  FRACTIONS. 

Definitions  and  Fundamental  Propositions 113 — 127 

To  reduce  a  Fraction  to  its  Lowest  Terms 123 — 129 

a  Fraction  to  an  Entire  or  Mixed  Quantity      .     .     .  130 
a  Mixed  Quantity  to  a  Fraction    .......  131 

Signs  of  Fractions 132 

To  reduce  Fractions  to  a  Common  Denominator 133 

the  Least  Common  Denominator  .  .  134 
To  reduce  a  Quantity  to  a  Fraction  with  a  given  Denominator  135 
To  convert  a  Fraction  "^"^  another  with  a  given  Denominator      .  136 

Addition  and  Subtraction  of  Fractions 137 — 138 

To  multiply  one  Fractional  Quantity  by  another 139 — 140 

To  divide  one  Fractional  Quantity  by  another 141 — 142 

To  reduce  a  Complex  Fraction  to  a  Simple  one       ....  143 
Resolution  of  Fractions  into  Series 144 

CHAPTER  IV— EQUATIONS  OF  THE  FIRST  DEGREE. 

Definitions  and  Elementary  Principles 145 — 152 

Transposition 153 

To  clear  an  Equation  of  Fractions 154 

Equations  of  the  First  Degree,  containing  one  Unknown  Quan- 
tity   155 

Questions  pro'lucing  Equations  of  the  First  Degree,  containing 

one  Unknown  Quantity  .     • 156 

Equations  of  the  First  Degree  containing  two  Unknown  Quau- 

titiea 157 

5 


PAGES. 

7—  24 

25—  26 

26—  31 

31—  33 

33-  39 

39—  43 

43—  46 

47—  48 

48—  50 

51 

53 

54—  65 

59—  63 

63—  68 

68—73 

74—  80 

80—  82 

83—  87 

87—  89 

92- 


95 


97—  99 
100-103 
103—107 
107—108 
108—109 


110—112 
112-113 
113—115 

115-119 

119—131 

132 


VI 


CONTENTS 


ARTICLES.  PAGES. 

Elimination — by  Substitution 158  132 

by  Comparison 159  133 

by  Addition  and  Subtraction 160  134 — 136 

Questions  producing  Equations  containing  two  Unknown  Quan- 
tities       IGl  136—142 

Equations  containing  three  or  more  Unknown  Quantities     .     .  162  143 — 140 

Questions  producing  Equations  containing  three  or  more  Un- 
known Quantities 163  147 — 150 

CHAPTER  V— SUPPLEMENT  TO  EQUATIONS  OF  THE  FIRST  DEGREE. 

Generalization — Formation  of  Rules — Examples 164 — 170  150 — 158 

Negative  Solutions 172  159 

Discus.sion  of  Problems 173  161 

Problem  of  the  Couriers 163 — 16:> 

Cases  of  Indetermination  and  Impossible  Problems    ....  174 — 177  165 — 167 

CHAPTER  VI— POWERS— ROOTS— RADICALS. 

Involution  or  Formation  of  Powers 178  168 

To  raise  a  Monomial  to  any  given  Power 179  168 

Polynomial  to  any  given  Power 181  170 

Fraction  to  any  Power 182  171 

Binomial  Theorem 183—186  171—176 

Extraction  of  the  Square  Root 176 

Square  Root  of  Numbers 1S7 — 190  176 — 179 

Fractions 101  179 

Perfect  and  Imperfect  Squares — Theorem 192  180 

Approximate  Square  Roots 193—194  181—183 

Square  Root  of  Monomials 195  183 — 184 

Polynomials 196  184^187 

Radicals  of  the  Second  Degree— Definitions 198  187 

Reduction 199  188 

Addition 200  189 

Subtraction 201  190 

Multiplication 202  191 

Division 203  192 

To  render  Rational,  the  Denominator  of  a  Fraction  containing 

Radicals 204  193 

Simple  Equations  containing  Radicals  of  the  Second  Degree     .  205  195 — 197 

CHAPTER  VII— EQUATIONS  OF  THE  SECOND  DEGREE. 

Definitions  and  Forms 206-208  197—193 

Incomplete  Equations  of  the  Second  Degree 209—210  198—200 

Questions  producing  Incomplete  Equations  of  the  Second  Degree,  211  200 — 201 

Complete  Equations  of  the  Second  Degree 212  202 

General  Rule  for  the  Solution  of  Complete  Equations  of  the  Sec- 
ond Degree      212  204—207 

Hindoo  Method  of  solving  Equations  of  the  Second  Degree        .  213  207 

Questions  producing  Complete  Equations  of  the  Second  Degree,  214  209 — 212 
Properties  of  the  Roots  of  a  Complete  Equation  of  the  Second 

Degree 215—218  213—217 

E(iuations  containing  two  Unknown  Quantities 219  217 — 220 

Questions  protlucing  Equations  of  the  Second  Degree,  routain- 

ing  two  Unknown  Quantities    .........  219  220 — 222 

CHAPTER  VIII— PROGRESSIONS  AND  PROPORTION. 

Arithmetical  Progression 220 — 225  222 — 227 

Geometrical  Progression 22C— 230  228—232 

Ratio 231—239  232—234 

Proportion  240—255  234—240 


RAY'S 

ALGEBRA 

PART  FIRST. 


INTELLECTUAL  EXERCISES. 


LESSON   I 


Note  to  Teachers. — All  the  exercises  in  the  following  lessons  can 
be  solved  in  the  same  manner  as  in  intellectual  arithmetic;  yet  the  instruc- 
tor should  require  the  pupils  to  perform  them  after  the  manner  here  indi- 
cated.    In  every  question  let  the  answer  be  verified. 

1.  I  have  15  cents,  which  I  wish  to  divide  between  William 
and  Daniel,  in  such  a  manner,  that  Daniel  shall  have  twice  as 
many  as  William  ;  what  number  must  I  give  to  each  ? 

If  I  give  William  a  certain  number,  and  Daniel  twice  that  num- 
ber, both  will  have  3  times  that  certain  number;  but  both  together 
are  to  have  15  cents  ;  hence,  3  times  a  certain  number  is  15. 

Now,  if  3  times  a  certain  number  is  15,  one-third  of  15,  or  5, 
must  be  the  number.  Hence,  AV^illiam  received  5  cents,  and  Dan- 
iel twice  5,  or  10  cents. 

If,  instead  of  a  certain,  number,  we  represent  the  number  of  cents 
William  is  to  receive,  by  x,  then  the  number  Daniel  is  to  receive 
will  be  represented  by  2x,  and  what  both  receive  will  be  repre 
sented  by  x  added  to  2x,  or  3x. 

If  3a3  is  equal  to  15, 
then  la;  or  X  is  equal  to  5. 

The  learner  will  see  that  the  two  methods  of  solving  this  ques- 
tion are  the  same  in  principle :  but  that  it  is  more  convenient  to 
represent  the  quantity  we  wish  to  find,  by  a  single  letter,  than  by 
one  or  more  words. 

In  the  same  manner,  let  the  learner  continue  to  use  the  letter  a: 
to  represent  the  smallest  of  the  required  numbers  in  the  following 
questions. 

7 


RAY'S    ALGEBRA,    PART    FIRST. 


Note. — x  is  read  x,  or  one  x,  and  is  the  same  as  \x.  2x  is  read  two  x, 
or  2  times  x.     3x  is  read  three  x,  or  3  times  x,  and  so  on. 

2.  What  number  added  to  itself  will  make  12? 

Let  X  represent  the  number  ;  then  x  added  to  x  makes  2x,  which 
is  equal  to  12  ;  hence  if  2x  is  equal  to  12,  one  ar,  which  is  the  half 
of  2x,  is  equal  to  the  half  of  12,  which  is  6. 

Verification. — 6  added  to  6  makes  12. 

3.  What  number  added  to  itself  will  make  16? 

If  X  represents  the  number,  what  will  represent  the  number 
added,  to  itself?  What  is  2x  equal  to?  If  2x  is  equal  to  16,  what 
is  X  equal  to  ? 

4.  What  number  added  to  itself  will  make  24  ? 

5.  Thomas  and  William  each  have  the  same  number  of  apples, 
and  they  both  together  have  20 ;  how  many  apples  has  each  ? 

6.  James  is  as  old  as  John,  and  the  sum  of  their  ages  is  22 
years  ;  what  is  the  age  of  each  ? 

7.  Each  of  two  men  is  to  receive  the  same  sum  of  money  for  a 
job  of  work,  and  they  both  together  receive  30  dollars ;  what  is 
the  share  of  each  ? 

8.  Daniel  had  18  cents  ;  after  spending  a  part  of  them,  he  found 
he  had  as  many  left  as  he  had  spent;  how  many  cents  had  he  spent? 

9.  A  pole  30  feet  high  was  broken  by  a  blast  of  wind  ;  the  part 
broken  off  was  equal  to  the  part  left  standing ;  what  was  the 
length  of  each  part  ? 

Instead  of  saying  x  added  to  x  is  equal  to  30,  it  is  more  conven- 
ient to  say  X  2)his  X  is  equal  to  30.  To  avoid  writing  the  word 
phis,  we  use  the  sign  +,  which  means  the  same,  and  is  called  the 
sign  of  addition.  Also,  instead  of  writing  the  word  equal,  we  use 
the  sign  ^=,  which  means  the  same,  and  is  called  the  sign  of 
equality. 

10.  John,  James,  and  Thomas,  are  each  to  have  equal  shares  of 
12  apples;  if  x  represents  John's  share,  what  will  represent  the 
share  of  James?  What  will  represent  the  share  of  Thomas? 
What  expression  will  represent  x-\-X'tx  more  briefly.  If  3x^=12, 
what  is  the  value  of  x  ?     AVhy  ? 

11.  The  sum  of  four  equal  numl)ers  is  equal  to  20 ;  if  a;  repre- 
sents one  of  the  numbers,  what  will  represent  each  of  the  others? 
What  will  represent  x-^x-\-x-rx,  more  briefly?  If4a;=20,  what 
is  X  equal  to  ?     Why  ? 

12.  What  is  x-{-x  equal  to?     Ans.  2x. 

13.  What  is  x-\-x^x  equal  to? 

14.  What  is  x-\-x-]rx-\-x  equal  to? 


INTELLECTUAL   EXERCISES. 


LESSON   II. 

1.  James  and  John  topjether  have  18  cents,  and  John  has  twice 
as  many  as  James ;  how  many  cents  has  each  ? 

If  a:  represents  the  number  of  cents  James  has,  what  will  repre- 
sent the  number  John  has  ?  What  will  represent  the  number  they 
both  have?     If  3a:  is  equal  to  18,  what  is  x  equal  to?     Why? 

Note. — If  the  pupil  does  not  readily  perceive  how  to  solve  a  question, 
let  the  instructor  ask  questions  similar  to  the  preceding. 

2.  A  travels  a  certain  distance  one  day,  and  twice  as  far  the 
next,  in  the  two  days  he  travels  36  miles ;  how  far  does  he  travel 
each  day  ? 

3.  The  sum  of  the  ages  of  Sarah  and  Jane  is  15  years,  and  the 
age  of  Jane  is  twice  that  of  Sarah ;  what  is  the  age  of  each  ? 

4.  The  sum  of  two  numbers  is  16,  and  the  larger  is  3  times  the 
smaller  ;  w*hat  are  the  numbers  ? 

5.  What  number  added  to  3  times  itself  will  make  20  ? 

6.  James  bought  a  lemon  and  an  orange  for  10  cents,  the  orange 
cost  four  times  as  much  as  the  lemon  ;  what  was  the  price  of  each? 

7.  In  a  store-room  containing  20  casks,  the  number  of  those 
that  are  full  is  four  times  the  number  of  those  that  are  empty; 
how  many  are  there  of  each  ? 

8.  In  a  flock  containing  28  sheep,  there  is  one  black  sheep  for 
each  six  w^hite  sheep ;  how  many  are  there  of  each  kind  ? 

9.  Two  pieces  of  iron  together  weigh  28  pounds,  and  the  hea- 
vier piece  weighs  three  times  as  much  as  the  lighter;  what  is  the 
weight  of  each  ? 

10.  William  and  Thomas  bought  a  foot-ball  for  30  cents,  and 
Thomas  paid  twice  as  much  as  William ;  w^hat  did  each  pay? 

1 1 .  Divide  35  into  two  parts,  such  that  one  shall  be  four  times 
the  other. 

12.  The  sum  of  the  ages  of  a  father  and  son  is  equal  to  35 
years,  and  the  age  of  the  father  is  six  times  that  of  his  son  ;  w^hat 
is  the  age  of  each  ? 

13.  There  are  two  numbers,  the  larger  of  which  is  equal  to  nine 
times  the  smaller,  and  their  sum  is  40 ;  what  are  the  numbers  ? 

14.  The  sum  of  tw^o  numbers  is  56,  and  the  larger  is  equal  to 
seven  times  the  smaller ;  what  are  the  numbers  ? 

15.  What  is  x-{-2x  equal  to? 

16.  What  is  x-\-Sx  equal  to? 

17.  What  is  x-\-4x  equal  to? 


10  RAY'S   ALGEBRA,    PART   FIRST. 


LESSON   III. 

1.  Three  boys  are  to  share  24  apples  between  them  ;  the  second 
is  to  have  twice  as  many  as  the  first,  and  the  third  three  times  as 
many  as  the  first.  If  x  represents  the  share  of  the  first,  what  will 
represent  the  share  of  the  second?  What  will  represent  the  share 
of  the  third?  What  is  the  sum  of  x-{-2x-\-'Sx'i  If  Gx  is  equal 
to  24,  what  is  the  value  of  x?  What  is  the  share  of  the  second? 
Of  the  third  ? 

Verification. — The  first  received  4,  the  second  twice  as 
many,  which  is  8,  and  the  third  three  times  the  first,  or  12 ;  and 
4  added  to  8  and  12,  make  24,  the  whole  number  to  be  divided. 

2.  There  are  three  numbers  whose  sum  is  30,  the  second  is 
equal  to  twice  the  first,  and  the  third  is  equal  to  three  times  the 
first ;  what  are  the  numbers  ? 

3.  There  are  three  numbers  whose  sum  is  21,  the  second  is 
equal  to  twice  the  first,  and  the  third  is  equal  to  twice  the  second. 
If  X  represents  the  first,  what  will  represent  the  second  ?  If  2x 
represents  the  second,  what  will  represent  the  third  ?  What  is 
the  sum  of  x-|-2x+4x?     What  are  the  numbers? 

4.  A  man  travels  63  miles  in  3  days ;  he  travels  twice  as  far 
the  second  day  as  the  first,  and  twice  as  far  the  third  day  as  the 
second ;  how  many  miles  does  he  travel  each  day  ? 

5.  John  had  40  chestnuts,  of  which  he  gave  to  his  brother  a 
certain  number,  and  to  his  sister  twice  as  many  as  to  his  brother ; 
after  this  he  had  as  many  left  as  he  had  given  to  his  brother;  how 
many  chestnuts  did  he  give  to  each  ? 

6.  A  farmer  bought  a  sheep,  a  cow,  and  a  horse,  for  60  dollars  ; 
the  cow  cost  three  times  as  much  as  the  sheep,  and  the  horse  twice 
as  much  as  the  cow  ;  what  was  the  cost  of  each  ? 

7.  James  had  30  cents ;  he  lost  a  certain  number ;  after  this 
he  gave  away  as  many  as  he  had  lost,  and  then  found  that  he  had 
three  times  as  many  remaining  as  he  had  given  away ;  how  many 
did  he  lose  ? 

8.  The  sum  of  three  numbers  is  36 ;  the  second  is  equal  to 
twice  the  first,  and  the  third  is  equal  to  three  times  the  second ; 
what  are  the  numbers? 

9.  John,  James,  and  William  together  have  50  cents  ;  John  has 
twice  as  many  as  James,  and  James  has  three  times  as  many  as 
AYilliam  ;  how  many  cents  has  each? 

10.  What  is  the  sum  of  x,  2x,  and  three  times  2x? 

11.  What  is  the  sum  of  twice  2x,  and  three  times  3a;? 


INTELLECTUAL    EXERCISES.  11 


LESSON   IV. 

1 .  If  1  lemon  costs  x  cents,  what  will  represent  the  cost  of  2 
lemons?     Of  3 ?    Of  4 ?     Of  5?    Of  6?    Of  7? 

2.  If  1  lemon  costs  2x  cents,  what  will  represent  the  cost  of  2 
lemons  ?     Of  3  ?     Of  4  ?     Of  5  ?     Of  6  ? 

3.  James  bought  a  certain  number  of  lemons  at  2  cents  a  piece, 
and  as  many  more  at  3  cents  a  piece,  all  for  25  cents ;  if  x  repre- 
sents the  number  of  lemons  at  2  cents,  what  will  represent  their 
cost  ?  What  will  represent  the  cost  of  the  lemons  at  3  cents  a 
piece  ?     How  many  lemons  at  each  price  did  he  buy  ? 

4.  Mary  bought  lemons  and  oranges,  of  each  an  equal  number ; 
the  lemons  cost  2,  and  the  oranges  3  cents  a  piece ;  the  cost  of  the 
whole  was  30  cents  ;  how  many  were  there  of  each  ? 

5.  Daniel  bought  an  equal  number  of  apples,  lemons,  and 
oranges  for  42  cents ;  each  apple  cost  1  cent,  each  lemon  2  cents, 
and  each  orange  3  cents ;  how  many  of  each  did  he  buy  ? 

6.  Thomas  bought  a  number  of  oranges  for  30  cents,  one-half 
of  them  at  2,  and  the  other  half  at  3  cents  each  ;  how  many 
oranges  did  he  buy  ?     Let  a;=  one-half  the  number. 

7.  Two  men  are  40  miles  apart ;  if  they  travel  toward  each 
other  at  the  rate  of  4  miles  an  hour  each,  in  how  many  hours  will 
they  meet? 

8.  Two  men  are  28  miles  asunder ;  if  they  travel  toward  each 
other,  the  first  at  the  rate  of  3,  and  the  second  at  the  rate  of  4 
miles  an  hour,  in  how  many  hours  will  they  meet? 

9.  Two  men  travel  toward  each  other,  at  the  same  rate  per 
hour,  from  two  places  whose  distance  apart  is  48  miles,  and 
they  meet  in  six  hours ;  how  many  miles  per  hour  does  each 
travel ? 

10.  Two  men  travel  toward  each  other,  the  first  going  twice  as 
fast  as  the  second,  and  they  meet  in  2  hours;  the  places  are  18 
miles  apart;  hoAv  many  miles  per  hour  does  each  travel? 

11.  James  bought  a  certain  number  of  lemons,  and  twice  as 
many  oranges,  for  40  cents ;  the  lemons  cost  2,  and  the  oranges 
3  cents  a  piece ;  how  many  were  there  of  each  ? 

12.  Two  men  travel  in  opposite  directions ;  the  first  travels 
three  times  as  many  miles  per  hour  as  the  second ;  at  the  end  of 
3  hours  they  are  36  miles  apart ;  hoAV  many  miles  per  hour  does 
each  travel  ? 

13.  A  cistern,  containing  100  gallons  of  water,  has  2  pipes  to 
empty  it ;  the  larger  discharges  four  times  as  many  gallons  per 


1^  KAY'S    ALGEBRA,    PART    FIRST. 


hour  as  the  smaller,  and  they  both  empty  it  in  2  hours ;  how  many 
gallons  per  hour  does  each  discharge  ? 

14.  A  grocer  sold  1  pound  of  coffee  and  2  pounds  of  tea  for  108 
cents,  and  the  price  of  a  pound  of  tea  was  four  times  that  of  a 
pound  of  coffee :  what  AA'^as.the  price  of  each  ? 

If  X  represents  the  price  of  a  pound  of  coffee,  what  will  repre- 
sent the  price  of  a  pound  of  tea  ?  What  will  represent  the  cost 
of  both  the  tea  and  coffee  ? 

15-  A  grocer  sold  1  pound  of  tea,  2  pounds  of  coffee,  and  3 
pounds  of  sugar,  for  65  cents  ;  the  price  of  a  pound  of  coffee  was 
twice  that  of  a  pound  of  sugar,  and  the  price  of  a  pound  of  tea 
Avas  three  times  that  of  a  pound  of  coffee.  Required  the  cost  of 
each  of  the  articles. 

If  a:  represents  the  price  of  a  pound  of  sugar,  what  will  repre- 
sent the  price  of  a  pound  of  coffee  ?  Of  a  pound  of  tea  ?  What 
will  represent  the  cost  of  the  whole  ? 


LESSON    V. 

1.  James  bought  2  apples  and  3  peaches,  for  16  cents;  the  price 
of  a  peach  was  twice  that  of  an  apple  ;  what  was  the  cost  of  each? 

If  X  represents  the  cost  of  an  apple,  what  will  represent  the 
cost  of  a  peach  ?  What  will  represent  the  cost  of  2  apples  ?  Of 
3  peaches  ?     Of  both  apples  and  peaches  ? 

2.  There  are  two  numbers,  the  larger  of  which  is  equal  to  twice 
the  smaller,  and  the  sum  of  the  larger  and  twice  the  smaller  is 
equal  to  28  ;  what  are  the  numbers  ? 

3.  Thomas  bought  5  apples  and  3  peaches  for  22  cents ;  each 
peach  cost  twice  as  much  as  an  apple ;  what  was  the  cost  of  each? 

4.  William  bought  2  oranges  and  5  lemons  for  27  cents ;  each 
orange  cost  twice  as  much  as  a  lemon  ;  what  was  the  cost  of 
each? 

5.  James  bought  an  equal  number  of  apples  and  peaches  for  21 
cents  ;  the  apples  cost  1  cent,  and  the  peaches  2  cents  each ;  how 
many  of  each  did  he  buy  ? 

6.  Thomas  bought  an  equal  number  of  peaches,  lemons,  and 
oranges,  for  45  cents ;  the  peaches  cost  2,  the  lemons  3,  and  the 
oranges  4  cents  a  piece ;  how  many  of  each  did  he  buy  ? 

7.  Daniel  bought  twice  as  many  apples  as  peaches  for  24  cents; 
each  apple  cost  2  cents,  and  each  peach  4  cents ;  how  many  of 
each  did  he  buy  ? 


INTELLECTUAL   EXERCISES.  13 

8.  A  farmer  bought  a  horse,  a  cow,  and  a  calf,  for  70  dollars ; 
the  COAV  cost  three  times  as  much  as  the  calf,  and  the  horse  twice 
as  much  as  the  cow ;  what  was  the  cost  of  each  ? 

9.  Susan  bought  an  apple,  a  lemon,  and  an  orange,  for  16  cents; 
the  lemon  cost  three  times  as  much  as  the  apple,  and  the  orange 
as  much  as  both  the  apple  and  the  lemon  :  what  was  the  cost  of 
each? 

10.  Fanny  bought  an  apple,  a  peach,  and  an  orange,  for  18 
cents ;  the  peach  cost  twice  as  much  as  the  apple,  and  the  orange 
twice  as  much  as  both  the  apple  and  the  peach ;  what  was  the 
cost  of  each  ? 


LESSON    VI. 

1.  James  bought  a  lemon  and  an  orange  ;  the  orange  cost  twice 
as  much  as  the  lemon,  and  the  difference  of  their  prices  was  2 
cents  ;  what  was  the  cost  of  "each  ? 

If  X  represent  the  cost  of  the  lemon,  Avhat  will  represent  the 
cost  of  the  orange  ?     What  is  2x  less  x  represented  by  ? 

2.  What  is  3a:  less  x  represented  by  ?  What  is  3a;  less  2x  repre- 
gented  by  ? 

What  is  4x  less  x  represented  by  ?  What  is  5a;  less  2x  repre- 
sented by  ? 

The  word  minus,  is  used  instead  of  less;  and  the  sign  — ,  for 
the  sake  of  brevity,  is  used  to  avoid  writing  the  Avord  minus. 

Thus,  if  we  wish  to  take  the  difference  between  3x  and  a;,  we 

may  say, 

Sx  less  X, 
or  3x  minus  x ;  which  may  be  written  3a; — x. 

When  the  sign  —  is  used,  it  is  to  be  read  minus. 

3.  Thomas  bought  a  lemon  and  an  orange ;  the  orange  cost 
three  times  as  much  as  the  lemon,  and  the  difference  of  their 
prices  was  4  cents  ;  what  was  the  price  of  each  ?  If  x  represents 
the  cost  of  the  lemon,  what  will  represent  the  cost  of  the  orange  ? 
What  is  3a; — x  represented  by  ? 

4.  In  a  school  containing  classes  in  Grammar,  Geography,  and 
Arithmetic,  there  are  three  times  as  many  studying  Geography  as 
Grammar,  and  twice  as  many  studying  Arithmetic  as  Geography ; 
there  are  10  more  in  the  class  in  Arithmetic  than  in  that  in  Grsim- 
mar ;  how  many  more  are  there  in  each  class  ?  If  a;  represents  the 
number  in  the  class  in  Grammar,  what  will  represent  the  number 


14  RAY'S   ALGEBRA,    PART   FIRST. 

in  the  class  in  Geography  ?     In  the  class  in  Arithmetic  ?     What 
is  6a: — X  represented  by  ?     What  is  it  equal  to  ? 

5.  The  age  of  Sarah  is  three  times  the  age  of  Jane,  and  the 
difference  of  their  ages  is  12  years ;  what  is  the  age  of  each? 

6.  The  difference  of  two  numbers  is  28,  and  the  greater  is  equal 
to  eight  times  the  less ;  what  are  the  numbers  ? 

7.  Daniel  has  four  times  as  many  cents  as  William,  and  Joseph 
has  twice  as  many  as  both  of  them ;  but  if  twice  the  number  of 
Daniel's  cents  be  taken  from  Joseph's,  the  remainder  is  only  16; 
how  many  cents  has  each  ? 

8.  Susan  bought  a  lemon,  an  orange,  and  a  pine-apple ;  the 
orange  cost  twice  as  much  as  the  lemon,  and  the  pine-apple  three 
times  as  much  as  both  the  lemon  and  the  orange ;  the  pine-apple 
cost  14  cents  more  than  the  orange ;  what  was  the  cost  of  each  ? 

9.  James  bought  1  lemon  and  2  oranges ;  an  orange  cost  twice 
as  much  as  a  lemon,  and  the  difference  between  the  cost  of  the 
oranges  and  the  lemon  was  6  cents  ;  what  was  the  cost  of  each  ? 

10.  Charles  bought  2  lemons  and  3  oranges ;  an  orange  cost 
twice  as  much  as  a  lemon,  and  the  difference  between  the  cost 
of  the  lemons  and  the  oranges  AA-as  8  cents ;  what  was  the  cost 
of  each? 

11.  A  man  bought  a  coav,  a  calf,  and  a  horse;  the  cow  cost 
twice  as  much  as  the  calf,  and  the  horse  twice  as  much  as  the 
cow ;  the  difference  between  the  price  of  the  horse  and  that  of  the 
calf  was  30  dollars  ;  what  was  the  cost  of  each  ? 

12.  There  are  three  numbers,  of  which  the  second  is  three 
times  the  first,  and  the  third  is  twice  as  much  as  both  the  first 
and  second,  while  the  difference  between  the  second  and  third 
is  10  ;  what  are  the  numbers? 


LESSON    VII. 

1.  James  and  John  together  have  11  cents,  and  John  has  3 
more  than  James  ;  how  many  has  each  ? 

If  James  has  x  cents,  then  John  has  a;+3,  and  they  both  have 
cc-fx-1-3,  or  2x'+3  cents  ;  hence,  2x+3  are  equal  to  11  :  hence,  if 
2x  and  3  are  equal  to  11,  2x  must  be  equal  to  1 1  less  3,  which  is 
equal  to  8  ;  then,  if  %x  is  equal  to  8,  onex,  or  x,  must  be  equal  to  4. 

2.  William  and  Daniel  together  have  9  apples,  and  Daniel  has 
one  more  than  William;  how  many  has  each?     If  x  represents 


INTELLECTUAL    EXERCISES.  15 

the  apples  William  has,  what  will  represent  the  apples  Daniel 
has  ?     What  will  represent  the  number  they  both  have  ? 

3.  In  a  class  containing  13  pupils,  there  are  three  more  boys 
than  girls  ;  how  many  are  there  of  each  ? 

4.  In  a  store-room  containing  40  barrels,  the  number  of  those 
that  are  empty  exceeds  the  number  filled  by  10;  how  many  are 
there  of  each  ? 

5.  In  a  flock  of  fifty  sheep,  the  number  of  those  that  are  white 
exceeds  the  number  that  are  black,  by  30  ;  how  many  are  there  of 
each  kind? 

6.  Two  men  together  can  earn  60  dollars  in  a  month,  but  one 
of  them  can  earn  10  dollars  more  than  the  other;  how  many 
dollars  can  each  earn  ? 

7.  The  sum  of  two  numbers  is  25,  and  the  larger  exceeds  the 
smaller  by  15  ;  what  are  the  numbers  ? 

8.  Sarah  and  Jane  bought  a  toy  for  25  cents,  of  which  Jane 
paid  5  cents  more  than  Sarah ;  how  much  did  each  pay  ? 

9.  The  difierence  between  two  numbers  is  4,  and  their  sum  is 
16;  what  are  the  numbers?  If  x  represents  the  smaller  number, 
what  will  represent  the  larger  ? 

10.  The  diifference  between  two  numbers  is  5,  and  their  sum  is 
35  ;  what  are  the  numbers  ?  . 


LESSON    VIII 


1 .  James  and  John  together  have  1 5  cents,  and  John  has  twice 
as  many  as  James,  and  3  more ;  how  many  has  each  ? 

If  X  represents  the  number  James  has,  then  2x+3  will  repre- 
sent the  number  John  has,  and  x+2x+3,  or  3x+3,  what  they 
both  have.  If  3x-|-3  is  equal  to  15,  then  3x-  must  be  equal  to  15 
less  3,  or  12  ;  hence  x  is  equal  to  4,  the  number  James  has  ;  then 
John  has  11. 

2.  William  bought  a  lemon  and  an  orange  for  7  cents ;  the 
orange  cost  twice  as  much  as  the  lemon  and  1  cent  more ;  what 
was  the  cost  of  each  ? 

3.  There  are  two  numbers  whose  sum  is  35 ;  the  second  is 
twice  the  first  and  5  more ;  what  are  the  numbers  ? 

4.  In  an  orchard  containing  apple-trees  and  cherry-trees,  the 
number  of  apple-trees  is  three  times  that  of  the  cherry-trees,  and 
7  more;  the  whole  number  of  trees  in  the  orchard  is  51  ;  how 
many  are  there  of  each  kind? 


16  KAY'S   ALGEBRA,    PART  FIRST. 

5.  A  farmer  bought  a  cow  and  a  calf,  for  13  dollars;  the  cow 
cost  three  times  as  much  as  the  calf,  and  1  dollar  more  ;  what  was 
the  cost  of  each  ? 

6.  William  and  Thomas  gave  50  cents  to  a  poor  woman;  Wil- 
liam gave  twice  as  many  as  Thomas,  and  5  cents  more ;  how  many 
cents  did  each  give  ? 

7.  Eliza  and  Jane  bought  a  doll  for  14  cents  ;  Eliza  paid  twice 
us  much  as  Jane,  and  2  cents  more ;  what  did  each  pay  ? 

8.  Divide  the  number  15  into  two  parts,  so  that  one  pai-t  shall 
exceed  the  other  by  3. 

9.  Divide  the  number  26  into  two  parts,  so  that  the  greater  part 
shall  be  5  more  than  twice  the  less  part. 

10.  The  sum  of  two  numbers  is  23,  and  the  greater  is  equal  to 
three  times  the  less,  and  3  more ;  what  are  the  numbers  ? 

11.  Two  numbers  added  together  make  40;  the  greater  is  5 
times  the  less,  and  4  more ;  what  are  the  numbers  ? 

12.  A  man  has  tw^o  flocks  of  sheep ;  the  larger  contains  six 
times  as  many  as  the  smaller,  and  5  more,  and  the  number  in 
both  is  82 ;  how  many  are  there  in  each? 


LESSON   IX. 

1.  James  has  as  many  cents  as  John,  and  2  more,  and  Thomas 
has  as  many  as  John,  and  3  more ;  they  all  have  26  cents ;  how 
many  has  each  ?  If  a;  represents  the  number  of  cents  John  has, 
what  will  represent  the  number  James  has  ?  The  number  Thomas 
has  ?     The  number  they  all  have  ? 

2.  James,  Thomas,  and  John,  went  out  to  gather  chestnuts ; 
Thomas  gathered  5  more  than  James,  and  John  3  more  than 
Thomas,  and  they  all  gathered  34;  how  many  did  each   gather? 

3.  A  father  distributed  25  cents  among  his  three  boys ;  to  the 
second  he  gave  2  more  than  to  the  first,  and  to  the  third,  3  more 
than  to  the  second ;  how  many  did  he  give  to  each  ? 

4.  Divide  the  number  19  into  three  parts,  so  that  the  first  may 
be  2  more  than  the  second,  and  the  third  twice  as  much  as  the 
second,  and  1  more. 

5.  Divide  13  apples  between  three  boys,  so  that  the  second  shall 
have  1  more  than  the  first,  and  the  third,  2  more  than  the  second. 

6.  A  peach,  a  lemon,  and  an  orange,  cost  15  cents ;  the  lemon 
cost  1  cent  more  than  twice  as  much  as  the  peach,  and  the  orange 


INTELLECTUAL   EXERCISES.  17 

2  cents  more  than  three  times  as  much  as  the  peach ;  how  many 
cents  did  each  cost  ? 

7.  Three  pieces  of  lead  together  weigh  47  pounds ;  the  second 
is  twice  the  weight  of  the  first,  and  the  third  weighs  7  pounds 
more  than  the  second ;  what  is  the  weight  of  each  piece? 

8.  The  sum  of  the  ages  of  Eliza,  Jane,  and  Sarah,  is  38  years , 
Jane  is  3  years  older  than  Eliza,  and  Sarah  is  2  years  older  than 
Jane  ;  what  are  their  ages  ? 

9.  A  father  has  three  sons,  each  of  whom  is  2  years  older  than 
his  next  younger  brother,  and  the  sum  of  their  ages  is  27  years ; 
what  is  the  age  of  each  ? 

10.  The  sum  of  three- numbers  is  29;  the  second  is  twice  the 
first  and  1  more,  and  the  third  is  equal  to  the  second,  and  2  more ; 
what  are  the  numbers  ? 

11.  A  man  bought  2  pounds  of  cofiee  and  1  pound  of  tea,  for 
50  cents ;  the  price  of  a  pound  of  tea  was  10  cents  more  than 
twice  the  price  of  a  pound  of  cofiee ;  what  did  each  cost  ? 

12.  A  man  bought  3  pounds  of  cofiee  and  1  pound  of  tea,  for  77 
cents ;  the  price  of  a  pound  of  tea  was  equal  to  the  price  of  2 
pounds  of  cofiee,  and  7  cents  more ;  what  was  the  price  of  each? 

13.  Says  A  to  B,  "  Good  morning,  master,  with  your  hundred 
geese."  Says  B,  "I  have  not  100;  but,  if  I  had  tAvice  as  many 
as  I  now  have,  and  20  more,  I  should  have  100."  How  many 
had  he  ? 


LESSON   X. 

1.  If  x-f  1  represent  a  certain  number,  what  will  represent 
twice  that  number  ?  Since  twice  x  is  2a;,  and  twice  1  is  2,  twice 
x+1,  will  be  represented  by  ^ZxAr'^' 

2.  What  is  3  times  x+1  ?     4  times  a;+l  ?     5  times  ar-fl  ? 

3.  If  a;+2  represent  a  certain  number,  what  will  represent 
twice  that  number  ?  2  times  x  is  2a:,  and  2  times  2  is  4,  hence, 
twice  ic-|-2  is  2x4-4. 

4.  What  is  3  times  a;+2?     4  times  a:+2  ?     5  times  x+2  ? 

5.  If  2x+l  represent  a  certain  number,  what  will  represent 
twice  that  number?  Twice  2x  is  4x,  and  twice  1  is  2,  hence, 
twice  2x4-1  is  4x4-2. 

6.  What  is  3  times  2x4-1?     4  times  2x4-1?     5  times  2x4-1? 

7.  What  is  2  times  3x4-2?     3  times  3x4-2?    4  times  3x4-2? 

8.  What  is  x,  x4-l,  and  x4-2  equal  to? 

2 


18  RAY'S   ALGEBRA,    PART    FIRST. 

9.  What  is  X,  x+l,  and  Sx+S  equal  to? 

10.  What  is  X,  a,-+3.  and  2a-+2  equal  to  ? 

11.  A  father  divided  15  cents  between  his  three  boys ;  giving 
to  the  second  1  more  than  to  the  first,  and  to  the  third  twice  as 
many  as  to  the  second ;  how  many  cents  did  each  receive  ? 

12.  The  sum  of  3  numbers  is  34  ;  the  second  is  1  more  than  the 
first,  and  the  third  is  3  times  the  second  ;  what  are  the  numbers? 

13.  Eliza,  Jane,  and  Sarah,  together  have  24  cents;  Jane  has 
twice  as  many  as  Eliza,  and  1  more,  and  Sarah  has  twice  as  many 
as  Jane  :  how  many  cents  has  each  ? 

14.  A  man  bought  1  pound  of  coffee  and  2  pounds  of  tea,  for  62 
cents ;  the  price  of  a  pound  of  tea  was  et[ual  to  that  of  2  pounds 
of  coffee,  and  1  cent  more ;  what  was  the  cost  of  each  ? 

15.  A  man  worked  three  days  for  10  dollars  ;  the  second  day  he 
earned  1  dollar  more  than  the  first,  and  the  third  day  as  much  as 
both  the  first  and  second ;  how  much  did  he  earn  each  day  ? 

16.  Three  boys  together  spent  43  cents;  the  second  spent  5 
cents  more  than  the  first,  and  the  third  twice  as  much  as  the 
second ;  how  many  cents  did  each  spend  ? 

17.  Divide  the  number  33  into  three  parts,  so  that  the  second 
shall  be  2  more  than  the  first,  and  the  third  equal  to  five  times  the 
second. 

18.  Three  men.  A,  B,  and  C,  have  40  dollars  between  them  ;  B 
has  twice  as  many  as  A,  and  1  dollar  more,  and  C  has  3  times  as 
many  as  B ;  how  many  dollars  has  each  ? 

19.  Divide  the  number  29  into  three  parts,  such  that  the  second 
shall  be  equal  to  the  first,  and  1  more,  and  the  third  equal  to  three 
times  the  second. 

20.  A  man  bought  3  pounds  of  sugar  and  2  pounds  of  coffee, 
for  41  cents ;  the  price  of  a  pound  of  coffee  was  3  cents  more 
than  that  of  a  pound  of  sugar ;  what  was  the  cost  of  each  ? 

21.  James  bought  2  lemons  and  3  oranges,  for  27  cents  ;  an 
orange  cost  twice  as  much  as  a  lemon,  and  1  cent  more ;  what 
was  the  cost  of  each  ? 

22.  An  apple,  a  peach,  and  2  pears,  cost  17  cents;  the  peach 
cost  1  cent  more  than  the  apple,  and  each  pear  twice  as  much  as 
the  peach ;  what  was  the  cost  of  each? 

23.  An  apple,  2  peaches,  and  3  pears,  cost  14  cents  ;  a  peach 
cost  1  cent  more  than  the  apple,  and  a  pear  1  cent  more  than  a 
peach ;  what  was  the  cost  of  each  ? 

24.  Two  pears,  3  lemons,  and  4  oranges,  cost  29  cents;  a 
lemon  cost  1  cent  more  than  a  pear,  and  an  orange  1  cent  mor© 
than  a  lemon  ;  what  was  the  cost  of  each  ? 


INTELLECTUAL   EXERCISES.  19 


LESSON   XI. 

1.  J<ames  has  4  cents,  and  John  has  1  cent  less  than  James; 
how  many  cents  has  John?  What  is  1  less  than  4?  What  is  2 
less  than  4  ? 

2.  If  X  represents  a  certain  number,  what  will  represent  1  less 
than  that  number?     Ans.  x — 1;  read  x  minus  1. 

3.  If  X  represents  a  certain  number,  what  will  represent  2  less 
than  that  number?    What  will  represent  3  less  than  that  number? 

4.  If  a  certain  number  less  1  is  equal  to  3,  what  is  the  number 
equal  to? 

5.  If  X — 1  is  equal  to  3,  what  is  x  equal  to? 

6.  If  2x— I  is  equal  to  5,  what  is  2x  equal  to?  If  2x  is  equal 
to  6,  what  is  x  equal  to? 

7.  If  3x — 2  is  equal  to  10,  what  is  3x  equal  to?  If  3x  is  equal 
to  12,  what  is  x  equal  to? 

8.  If  5x — 3  is  equal  to  17,  what  is  5x  equal  to?  If  5x  is  equal 
to  20,  w^hat  is  x  equal  to  ? 

9.  James  and  John  together  have  17  cents,  and  James  has  3 
cents  less  than  John  ;  how  many  has  each  ? 

If  X  represents  the  number  of  cents  James  has,  what  will  repre- 
sent the  number  John  has  ?  What  is  x  and  x — 3  equal  to  ?  If 
2x— 3  is  equal  to  17,  what  is  2x equal  to?  If  2x  is  equal  to  20, 
what  is  X  equal  to  ? 

10.  Divide  the  number  17  into  two  parts,  so  that  one  shall  be 
5  less  than  the  other. 

11.  An  orange  and  a  lemon  together  cost  8  cents,  and  the  lemon 
cost  two  cents  less  than  the  orange  ;  what  was  the  cost  of  each  ? 

12.  The  sum  of  two  numbers  is  20,  and  the  smaller  is  4  less 
than  the  greater  ;  what  are  the  numbers  ? 

13.  William  and  Daniel  together  have  20  cents,  and  Daniel  has 
twice  as  many  as  William,  wanting  1  cent;  how  many  cents  has 
each? 

14.  The  sum  of  two  numbers  is  24,  and  the  larger  is  twice  the 
smaller,  wanting  3  ;  what  are  the  numbers  ? 

15.  In  a  basket  containing  25  apples  and  peaches,  if  5  be  sub- 
tracted from  twice  the  number  of  apples,  it  will  give  the  number 
of  peaches ;  how  many  are  there  of  each  ? 

16.  The  sum  of  two  numbers  is  25,  and  the  greater  is  equal  to 
3  times  the  smaller,  wanting  7;  what  are  the  numbers  ? 

17.  A  school  contains  37  pupils,  the  number  of  boys  is  3  times 
the  number  of  girls,  wanting  3  ;  what  is  the  number  of  each  ? 


20  RAY'S   ALGEBRA,    PART   FIRST. 

18.  A  cow,  a  calf,  and  a  sheep,  cost  28  dollars  ;  the  sheep  cost 
2  dollars  less  than  the  calf,  and  tJie  cow  cost  4  times  as  much  as 
the  calf;  what  was  the  cost  of  each? 


LESSON   XII. 

1 .  What  number  is  that,  to  which  if  3  be  added,  the  number 
will  be  doubled  ?  If  ic  represents  the  number,  what  will  a;+3  be 
equal  to  ? 

Since  2x  is  equal  to  x-f-3,  it  is  plain  that  x  is  equal  to  3. 

2.  What  number  is  that,  to  w^hich  if  5  be  added,  the  number 
will  be  doubled  ? 

3.  What  number  is  that,  to  which  if  4  be  added,  the  sum  will  be 
3  times  the  number  ?  If  a;  represents  the  number,  x-\-4  will  be 
equal  to  3x ;  but  if  3a;  is  equal  to  x-\-4,  it  is  plain  that  2x  is  equal 
to  4,  and  that  x  is  equal  to  2. 

4.  What  number  is  that,  to  which  if  9  be  added,  the  sum  will 
be  4  times  the  number?  If  x  represents  the  number,  what  will 
x-\-9  be  equal  to  ?  If  4a:  is  equal  a;H-9,  it  is  plain  that  3x  is 
equal  to  9,  and  that  x  is  equal  to  3. 

5.  What  number  is  that,  to  which  if  15  be  added,  the  sum  will 
be  four  times  the  number  ? 

6.  There  are  10  years  difference  between  the  ages  of  two 
brothers,  and  the  age  of  the  elder  is  3  times  that  of  the  younger  ; 
what  is  the  age  of  each  ? 

7.  James  says  to  John,  "  I  have  4  times  as  many  apples  as  you 
have  ;  but  if  you  had  9  apples  more  than  you  now  have,  we  would 
then  each  have  an  equal  number."     How  many  has  each? 

8.  The  difference  of  two  numbers  is  20,  and  the  greater  is  5 
times  the  smaller  ;  what  are  the  numbers  ? 

9.  The  age  of  Eliza  exceeds  that  of  Jane  16  years,  while  the 
age  of  the  former  is  five  times  that  of  the  latter ;  what  are  their 
ages  ? 

10.  James  bought  a  book  and  a  toy ;  the  book  cost  six  times  as 
much  as  the  toy,  and  the  difference  of  their  prices  was  20  cents ; 
how  much  did  he  pay  for  each  ? 

11.  The  difference  between  the  age  of  a  father  and  that  of  his 
son,  is  30  years,  and  the  age  of  the  father  is  seven  times  the  age 
of  the  son  ;  what  are  their  ages  ? 

,12.  What  number  is  that,  to  which  if  32  be  added,  the  sum  will 
be  equal  to  nine  times  the  number  itself? 


INTELLECTUAL   EXERCISES.  21 


13.  What  number  is  that  which  is  6  less  than  3  times  the 
number  itself? 

14.  James  is  12  years  younger  than  John;  but  John  is  only 
four  times  the  age  of  James ;  what  are  their  ages  ? 

1 5.  What  number  is  that,  to  the  double  of  which,  if  8  be  added, 
the  sum  will  be  equal  to  4  times  the  number  ? 

In  this  case,  if  x  represents  the  number,  4a:  is  equal  to  2x+8  ; 
hence  2x  must  be  equal  to  8,  and  x  equal  to  4. 

16.  What  is  the  value  of  x,  when  5x  is  equal  to  3x+6? 

17.  What  is  the  value  of  x,  when  5x  is  equal  to  2x-|-15? 

18.  What  is  the  value  of  x,  when  8x  is  equal  to  3x+15? 

19.  What  is  the  value  of  x,  when  lOx  is  equal  to  4x+24? 

20.  What  number  is  that,  to  the  double  of  which,  if  21  be 
added,  the  sum  will  be  five  times  the  number? 

21.  If  Daniel's  age  be  multiplied  by  4,  and  30  added  to  the 
product,  the  sum  will  be  6  times  his  age ;  what  is  his  age  ? 

22.  What  number  added  to  twice  itself  and  32  more,  will  make 
a  sum  equal  to  7  times  the  number  ? 

23.  What  number  added  to  itself  and  40  more,  will  make  a  sum 
equal  to  10  times  the  number? 

24.  A  father  gave  his  son  3  times  as  many  cents  as  he  then 
had,  his  uncle  then  gave  him  40  cents,  when  he  found  he  had  9 
times  as  many  as  at  first ;  how  many  had  he  at  first  ? 


LESSON   XIII. 

1.  What  number  is  that  which  being  increased  by  5,  and  then 
doubled,  the  sum  will  be  equal  to  three  times  the  number? 

In  this  example  let  x  represent  the  number,  then  x+5  doubled, 
will  be  2x+10,  which  is  equal  to  3x;  hence  x  is  equal  to  10. 

2.  Sarah  is  2  years  older  than  Jane,  and  twice  Sarah's  age  is 
equal  to  three  times  the  age  of  Jane ;  what  is  the  age  af  each  ? 

3.  William  has  8  cents  more  than  Daniel,  and  three  times  Wil- 
liam's money  is  equal  to  5  times  that  of  Daniel ;  how  many  cents 
has  each? 

4.  Three  pounds  of  coffee  cost  as  much  as  5  pounds  of  sugar, 
and  1  pound  of  coffee  cost  6  cents  more  than  1  pound  of  sugar ; 
what  is  the  price  of  a  pound  of  each  ? 

5.  A  farmer  bought  2  hogs  and  7  sheep  ;  a  hog  cost  5  dollars 
more  than  a  sheep,  while  the  hogs  and  sheep  both  cost  the  same 
sum  :  what  was  the  cost  of  each  ? 


22  RAY»S   ALGEBRA,    PART    FIRST. 


6.  William  bought  3  oranges  and  5  lemons ;  an  orange  cost  2 
cents  more  than  a  lemon,  while  the  oranges  and  the  lemons  each 
cost  the  same  sum  ;  what  was  the  cost  of  each  ? 

7.  William  has  10  cents  more  than  Daniel;  but  7  times  Dan- 
iel's money  is  equal  to  twice  that  of  William ;  how  many  cents 
has  each? 

8.  The  greater  of  2  numbers  exceeds  the  less  by  14;  and  3 
times  the  greater  is  equal  to  10  times  the  less;  what  are  the 
numbers  ? 

9.  Moses  is  16  years  younger  than  his  brother  Joseph ;  but  3 
times  the  age  of  Joseph  is  equal  to  5  times  that  of  Moses ;  what 
are  their  ages  ? 

10.  The  difference  between  the  ages  of  a  man  and  his  wife  is  7 
years ;  and  6  times  the  age  of  the  man  is  equal  to  8  times  the  age 
of  his  wife  ;  what  are  their  ages  ? 


LESSON    XIV. 

1.  If  a:  represents  a  certain  number,  what  will  represent  one 
half  the  number  ? 

To  divide  a  number,  we  draw  a  line  beneath  it,  under  which  we 

place  the  divisor ;  thus,  to  divide  1  by  2,  it  is  written  ^,  which   is 
read  one  lialf,  or  one  divided  by  two.     In  the  same  manner,  one  half 

X 

of  X  would  be  written  thus,  ^;  which  may  be  read  one  half  of  x,  or 
X  divided  by  2. 

In  a  similar  manner,  one  third  of  x  is  written  ^  ;   two  thirds  of 

2x  '^ 

X  is  written  -^. 
o 

X 

2.  If  ^  is  equal  to  4,  what  is  x  equal  to  ? 


X 

3.  If  o  is  equal  to  5,  w^hat  is  x  equal 
o 


to? 


2x 
4.  If  "o"  is  equal  to  8,  what  is  x  equal  to  ?     If  hco  thirds  of  x  is 

equal  to  8,  one  third  of  x  is  equal  to  one  half  of  8,  or  4  (since  one 
half  of  two  thirds  is  one  third);  and  if  one  third  of  x  is  equal  to  4, 
X  is  equal  to  three  times  4,  or  12. 

Or  thus:  if  2x  divided  by  3  is  equal  to  8,  2x  must  be  equal  to 
3  times  8,  or  24 ;  and  if  2x  is  equal  to  24,  x  is  equal  to  one  half 
of  24,  or  12. 


INTELLECTUAL  EXERCISES.  23 

Either  of  these  methods  may  be  used  in  finding  the  value  of  x 
in  similar  expressions. 

5.  If  -^  is  equal  to  9,  what  is  x  equal  to  ? 

6.  If  -^  is  equal  to  10,  what  is  x  equal  to  ? 

Ix  . 

7.  If  Y"!  is  equal  to  14,  Avhat  is  x  equal  to? 

3x 

8.  If -o"  is  equal  to  9,  what  is  x  equal  to? 

4x 

9.  If  -K-  is  equal  to  12,  what  is  x  equal  to? 

5x 

10.  If -o"  is  equal  to  20,  what  is  x  equal  to? 

7x 

11.  If  -F-  is  equal  to  14,  what  is  x  equal  to? 

9a; 

12.  If -=r  is  equal  to  18,  what  is  x  equal  to? 

13.  What  is  the  sum  of  x  and  ^  ?  or  of  x-\-^  ? 

Since  x  is  equal  to  -^,  we  have  a^+s  equal  to  -9+9*  which  is 
3x 


equal  to  -^ 


14.  AVhat  will  represent  the  sum  of  2x  and  ^,  or  of  2x-+^? 

15.  What  will  represent  the  sum  of  a;+o? 

2a:  x 

16.  What  will  represent  the  sum  of  a:+-^?     Of  2a:+q^ 

X  3a*  4a: 

17.  What  is  the  sum  of  a:+^?     Of  x-^-^'i     Of  2x+-^? 

2r  3a'  3a: 

18.  What  is  the  sum  of  x+y?     Ofa;+-g?     Of  2x+y? 

19.  There  is  a  certain  number,  to  which  if  the  half  of  itself  be 
lidded,  the  sum  will  be  15 ;  what  is  the  number? 

20.  William  has  half  as  many  cents  as  Daniel,  and  they  both 
together  have  2 1  ;  how  many  cents  has  each  ? 

21.  The  age  of  Mary  is  one  third  that  of  Jane,  and  the  sum  of 
their  ages  is  24  years  ;  what  is  the  age  of  each? 

22.  A  pasture  contains  44  sheep  and  cows  ;  the  number  of  cowa 
is  one  third  the  number  of  sheep  ;  how  many  are  there  of  each  ? 

23.  The  sum  of  the  ages  of  Ruth  and  Eliza  is  24  years ;  while 
the  age  of  the  former  is  three  fifths  of  that  of  the  latter ;  what  is 
the  age  of  each? 


24  RAY'S   ALGEBRA,    PART   FIRST. 

24.  James  and  John  together  have  18  cents,  and  John  has  four 
fifths  as  many  as  James  ;  how  many  has  each  ? 

25.  Two  phxces,  A  and  C,  are  40  miles  apart ;  between  them  is 
ii  village  which  is  two  thirds  as  far  from  C  as  it  is  from  A ;  what 
is  its  distance  from  each  of  the  places  ? 

26.  The  sum  of  two  numbers  is  21,  and  the  smaller  number  is 
three  fourths  of  the  larger  ;  what  are  the  numbers? 

27.  Thomas  and  Charles  have  35  cents,  and  Charles  has  half 
as  many  more  cents  as  Thomas  ;  how  many  cents  has  each  ? 

28.  The  double  of  a  certain  number,  increased  by  one  third  of 
itself,  is  equal  to  21  ;  what  is  the  number? 

29.  William,  James,  and  Robert,  together,  have  33  cents ;  James 
ha^  twice  as  many  as  William,  and  Robert  has  one  third  as  many 
as  James  ;  how  many  cents  has  each  ? 

30.  What  number  is  that,  which  being  increased  by  its  half  and 
its  fourth,  equals  21  ? 

31.  What  number  is  that,  which  being  increased  by  its  half,  its 
fourth,  and  4  more,  equals  25  ? 

32.  A  boy,  being  asked  how  much  money  he  had,  replied,  that 
if  one  half  and  one  third  of  his  money,  and  9  cents  more,  were 
added  to  it,  the  sum  would  be  20  cents  ;  how  much  money  had  he  ? 

33.  There  are  three  numbers,  whose  sum  is  44 ;  the  second  is 
equal  to  one  third  of  the  first,  and  the  third  is  equal  to  the  second 
and  twice  the  first :  what  are  the  numbers  ? 

34.  There  are  four  towns  in  the  order  of  the  letters.  A,  B,  C, 
and  D ;  the  distance  from  B  to  C  is  one  fifth  of  the  distance  from 
A  to  B,  and  the  distance  from  C  to  D  is  equal  to  twice  the  dis- 
tance from  A  to  C  ;  the  whole  distance  from  A  to  D  is  72  miles. 
Required  the  distance  from  A  to  B,  from  B  to  C,  and  from  C  to  D. 

35.  What  number  is  that,  to  which,  if  its  half,  its  fourth,  and  26 
more  be  added,  the  sum  will  be  equal  to  5  times  the  number  ? 

36.  There  is  a  fish  whose  head  is  6  inches  long,  and  the  tail  is 
as  long  as  the  head  and  half  the  body,  and  the  body  is  as  long  as 
the  head  and  tail ;  what  is  the  length  of  the  whole  fish  ? 

37.  A  gentleman  being  asked  his  age,  replied,  "  If  to  my  ago 
you  add  its  half,  its  third,  and  28  years,  the  sum  will  be  equal  to 
three  times  my  age."     Required  his  age. 

^^^  The  preceding  exercises  will  serve  to  give  the  learner  some  idea  of 
the  nature  of  Algebra,  and  of  the  manner  in  which  it  may  be  applied  to  tho 
solution  of  problems.  We  shall  now  proceed  to  consider  the  subject  in  a 
regular  and  scientific  manner. 


ELEMENTS  OF  ALGEBRA. 


CHAPTER  I. 

PRELIMIIVARY    DEFINITIONS    AND    PRINCIPLES. 

Note  to  Teachers. — In  general,  the  Introduction,  embracing  Ar- 
ticles 1  to  15,  need  not  be  thoroughly  studied  until  the  pupil  reviews  the 
book. 

Article  1.  In  Algebra,  numbers  and  quantities  are  represented 
by  symbols.     These  symbols  are  the  letters  of  the  alphabet. 

Art.  2.  Quantity  is  anything  that  is  capable  of  increase  or 
decrease ;  such  as  numbers,  lines,  space,  time,  motion,  &c. 

Art.  3.  Quantity  is  called  magnitude,  when  presented  or  con- 
sidered in  an  undivided  form,  such  as  a  quantity  of  water. 

Art.  4.  Quantity  is  called  multitude,  when  it  is  made  up  of  indi- 
vidual and  distinct  parts,  such  as  three  cents,  which  is  a  quantity 
composed  of  three  single  cents. 

Art.  5.  One  of  the  single  parts  of  which  a  quantity  of  multi- 
tude is  composed,  is  called  the  unit  of  quantity,  or  measui'ing  unit; 
thus,  one  cent  is  the  measuring  unit  of  the  quantity  three  cents. 
The  value  or  measure  of  every  quantity,  is  the  number  of  times  it 
contains  its  measuring  unit. 

Art.  6.  In  quantities  of  magnitude,  where  there  is  no  natural 
unit,  it  is  necessary  to  fix  upon  an  artificial  unit,  as  a  standard  of 
measure  ;  and  then  to  find  the  value  of  the  quantity,  we  must 
ascertain  how  often  it  contains  its  unit  of  measure.  Thus,  to 
measure  the  length  of  a  line,  we  take  a  certain  assumed  distance 
called  a  foot,  and  applying  it  a  certain  number  of  times,  say  five, 
we  ascertain  that  the  line  is  five  feet  long  ;  in  this  case,  one  foot 
is  the  vnit  of  measure. 

Art.  7.  The  numerical  value  of  any  quantity,  is  the  number  that 
expresses  how  many  times  it  contains  its  unit  of  measure.  Thus, 
in  the  preceding  example,  the  line  being  5  feet  long,  its  numerical 

Review. — 1.  How  are  numbers  and  quantities  represented  in  Algebra? 
"What  are  symbols  ?  2.  What  is  a  quantity  ?  3.  When  is  quantity  called 
magnitude?  4.  When  is  quantity  called  multitude  ?  5.  What  is  the  unit 
of  quantity  ?  6.  How  is  the  value  of  a  quantity  ascertained,  when  there  is 
no  natural  unit?     7.  What  is  the  numerical  value  of  any  quantity? 

3  25 


2G  RAY'S    ALGEBRA,    PART  FIRST. 

value  is  5.  The  same  quantity  may  have  diflferent  numerical 
values,  according  to  the  unit  of  measure  that  is  assumed. 

Art.  8.  A  unit  is  a  single  or  whole  thing  of  an  order  or  kind. 

Art.  9.  Number  is  an  expression  denoting  a  unit,  or  a  collec- 
tion of  unit:^     Numbers  are  either  abstract  or  concrete. 

Art.  10.  An  abstract  number  denotes  hovr  many  times  a  unit  is 
to  be  taken.  A  concrete,  or  applicate  number,  denotes  the  units 
that  are  taken. 

Thus,  4  feet  is  a  concrete  number ;  while  4  is  an  abstract  num- 
ber, "which  merely  shows  the  number  of  units  that  are  taken.  A 
concrete  number  may  be  defined  to  be  the  product  of  the  unit  of 
measure  by  the  corresponding  abstract  number.  Thus,  6  dollars 
are  equal  to  1  dollar  multiplied  by  6,  or  1  dollar  taken  6  times. 

Art.  11,  In  Algebra,  quantities  are  represented  by  numbers 
and  letters  ;  the  letters  used,  stand  for  numbers. 

Art.  12.  There  are  two  kinds  of  questions  in  Algebra,  theorems 
and  problems. 

Art.  13.  In  a  theorem,  it  is  required  to  demonstrate  some  rela- 
tion or  property  of  numbers,  or  abstract  quantities. 

Art.  14.  In  m  problem,  it  is  required  to  find  the  value  of  some 
unknown  number  or  quantity,  by  means  of  certain  given  relations 
existing  between  it  and  others,  which  are  known. 

Art.  15.  Algebra  is  a  general  method  of  solving  problems  and 
demonstrating  theorems,  by  means  of  figures,  letters,  and  signs. 
The  letters  and  signs  are  sometimes  called  symbols. 

DEFIIVITIOIV   OF  TERMS,   AND  EXPLANATIOIV   OF  SIGNS. 

Art.  16.  Knoicn  quantities  are  those  Avhose  numerical  values 
are  given,  or  supposed  to  be  known :  unknown  quantities  are  those 
whose  numerical  values  are  not  known. 

Art.  lY.  Known  quantities  are  generally  represented  by  the 
first  letters  of  the  alphabet,  as  a,  b,  c,  &c.;  and  unkno^^^l  quantities 
by  the  last  letters,  as  x,  y,  z. 

Art.  18.  The  following  are  the  principal  signs  used  in  Algebra: 

=,    +,    —      X,    -7-,     (    ),     >,     V- 
Each  of  these  signs   is  the   representative   of  certain  words ; 

Review. — 8.  What  is  a  unit?  9.  What  is  number?  10.  What  does 
an  abstract  number  denote?  What  does  a  concrete  number  denote?  11.  What 
do  the  letters  used  in  Algebra  represent?  12.  How  many  kinds  of  ques- 
tions are  there  in  Algebra?  What  are  they?  13.  What  is  a  theorem? 
14.  What  is  a  problem?  15.  What  is  Algebra?  16.  What  are  known 
quantities  ?  What  are  unknown  quantities  ?  17.  By  what  are  known  quan- 
tities represented  ?  By  what  are  unknown  quantities  represented  ?  18.  Writo 
on  a  slate,  or  a  blackboard,  the  principal  signs  used  in  Algebra.  What  do 
the  signs  represent?    For  what  purpose  are  they  used  ? 


DEFINITIONS    AND    NOTATION.  27 


they  are  used  for  the  purpose  of  expressing  the  various  operations, 
in  the  most  clear  and  brief  manner. 

Art.  19.  The  sign  of  equality,  =,  is  read  equal  to.  It  denotes  that 
the  quantities  between  which  it  is  placed  are  equal  to  each  other. 
Thus,  a=:3,  denotes  that  the  quantity  represented  by  a  is  equal  to  3. 

Art.  20.  The  sign  of  addition,  -\-,  is  readj?Z?«5.  It  denotes  that 
the  quantity  to  which  it  is  prefixed,  is  to  be  added  to  some  other 
quantity. 

Thus,  a-\-h  denotes  that  h  is  to  be  added  to  a.  If  a=2  and 
6=3,  then  a+5=:2+3,  which  are  =5. 

Art.  21.  The  sign  oi  suhti'action,  — ,  is  read  minus.  It  denotes 
that  the  quantity  to  which  it  is  prefixed  is  to  be  subtracted.  Thus, 
a — h  denotes  that  h  is  to  be  subtracted  from  a.  If  a=^5  and  6=3, 
then  5—3=2. 

Art.  22.  The  signs  +  and  —  are  called  the  signs ;  the  former 
is  called  the  'positive,  and  the  latter  the  negative  sign ;  they  are 
said  to  be  contrary  or  opposite. 

Art.  23.  Every  quantity  is  supposed  to  be  preceded  by  one  or  the 
other  of  these  signs.  Quantities  having  the  positive  sign  are  called 
positive;  and  those  having  the  negative  sign  are  called  negative. 
When  a  quantity  has  no  sign  prefixed  to  it,  it  is  considered  positive. 

Art.  24.  Quantities  having  the  same  sign  are  said  to  have  like 
signs ;  those  having  difierent  signs  are  said  to  have  unlike  signs. 
Thus,  -{-a  and  -j-6,  or  — a  and  — h  have  like  signs;  while  +c  and 
— d  have  unlike  signs. 

Art,  25.  The  sign  of  midtiplication,  Xj  is  read  into,  or  multi^ 
plied  by.  It  denotes  that  the  quantities  between  which  it  is  placed, 
are  to  be  multiplied  together. 

A  dot  or  point  is  sometimes  used  instead  of  the  sign  X«  Thus, 
aX6  and  a.b,  both  mean  that,  b  is  to  be  multiplied  by  a.  The  dot 
is  not  used  to  denote  the  multiplication  of  figures,  because  it  is 
used  to  separate  whole  numbers  and  decimals. 

The  product  of  two  or  more  letters  is  generally  denoted  by 
writing  them  in  close  succession.  Thus,  «6  denotes  the  same  as 
«X6>  or  a.b ;  and  abc  means  the  same  as  ay^by^c,  or  a.b.c. 

R  E  V I E  w. — 19.  How  is  tho  sign  of  equality,  =,  read  ?  What  does  it  de- 
note? 20.  How  is  the  sign -j- read  ?  What  does  it  denote  ?  21.  How  is 
tho  sign  —  read  ?  What  does  it  denote  ?  22.  What  are  the  signs  plus  and 
minus  called,  by  way  of  distinction  ?  Which  is  positive,  and  which  nega- 
tive ?  23.  When  quantities  are  preceded  by  the  sign  plus,  what  are  they 
said  to  be  ?  By  the  sign  minus  ?  When  a  quantity  has  no  sign  prefixed, 
what  sign  is  understood  ?  24.  When  do  quantities  have  like  signs  ?  When 
unlike  signs  ?  25.  How  is  the  sign  X  read,  and  what  does  it  denote  ?  What 
other  methods  are  there  of  representing  multiplication,  besides  tho  sign  X  ? 


28 

Art.  26.  Quantities  that  are  to  be  multiplied  together,  are  called 
factors.  The  continued  product  of  several  factors,  means  that  the 
product  of  the  first  and  second  is  to  be  multiplied  by  the  third,  this 
product  by  the  fourth,  and  so  on.  Thus,  the  continued  product  of 
a,  6,  and  c,  is  expressed  by  aX^Xc,  or  abc. 

If  a=2,  &=3,  and  c=5,  then  «6c==2X3Xo=6X5=30. 

Art.  2'7«  The  sign  of  division,  -r-,  is  read  divided  by.  It  denotes 
that  the  quantity  preceding  it  is  to  be  divided  by  that  following  it. 
The  division  of  two  quantities  is  more  frequently  represented,  by 
placing  the  dividend  as  the  numerator,  and  the  divisor  as  the  de- 
nominator of  a  fraction.     Thus,  a-r-h,  or  -,  means,  that  a  is  to 

be  divided  bv  h.    If  a=12  and  6=3,  then  a-7-6=12-7-3=4;  or 
a     12     . 

Division  is  also  represented  thus,  a\h,  where  a  denotes  the 
dividend,  and  h  the  divisor. 

Art.  28.  The  sign  >,  is  called  the  sign  of  ineqiiality.  It  de- 
notes that  one  of  the  two  quantities  between  which  it  is  placed,  is 
greater  than  the  other,  the  opening  of  the  sign  being  turned 
towards  the  greater  quantity. 

Thus,  a>6  denotes  that  a  is  greater  than  b.  It  is  read,  a  greater 
than  b.     If  a=5,  and  6=3,  then  5>3. 

Also,  c<Cd  denotes  that  c  is  less  than  d.     It  is  read,  c  less  than  d. 

If  c=4  and  d=^7,  then  4<7. 

Art.  29.  The  sign  oo,  denotes  a  quantity  greater  than  any  that 
can  be  assigned ;  that  is,  a  quantity  indefinitely  great,  or  infinity. 

Art.  30.  The  numeral  coefficient  of  a  quantity  is  a  number  pre- 
fixed to  it,  to  show  how  often  the  quantity  is  to  be  taken.  Thus, 
if  the  quantity  represented  by  a  is  to  be  added  to  itself  several 
times,  as  a-\-a-{-a-\-a,  we  write  it  but  once,  and  place  a  number 
before  it,  to  show  how  often  it  is  taken. 

Thus,  a-\-a^a-\-a=^4:a  ;  and  ax-\-(ix-\-ax^=2ax. 

Art.  31.  The  literal  coefficient  of  a  quantity,  is  a  quantity  by 
which  it  is  multiplied.  Thus,  in  the  quantity  ay,  a  may  be  consid- 
ered the  coefficient  of  y,  or  y  may  be  considered  the  coefficient  of  a. 
The  literal  coefficient  is  generally  regarded  as  a  known  quantity. 

Review. — 26.  What  are  factors?  How  many  factors  in  a?  In  ahl 
In  ahel  In  bahcl  27.  How  is  the  sign  -f-  read,  and  what  does  it  denote? 
What  other  methods  are  there  of  representing  the  division  of  two  quantities  ? 
28.  What  does  the  sign  of  inequality,  ^,  denote  ?  Which  quantity  is  placed 
at  the  opening?  29.  What  does  the  sign  00  denote?  30.  AVhat  is  the  nu- 
meral coefficient  of  a  quantity  ?  How  often  is  ax  taken  in  the  expression 
3aa??     In  5ax?     In^axl     31.  What  is  the  literal  coefficient  of  a  quantity  ? 


DEFINITIONS   AND   NOTATION.  29 

Art.  32.  The  coefficient  of  a  quantity  may  consist  of  a  number, 
and  also  of  a  literal  part.  Thus,  in  the  quantity  5ax,  5a  may  be  re* 
garded  as  the  coefficient  of  x.    If  a=2,  then  5a=10,  and  5ax=l0x. 

When  no  numeral  coefficient  is  prefixed  to  a  quantity,  its  coef- 
ficient is  understood  to  be  unity.  Thus,  a  is  the  same  as  la,  and 
hx  is  the  same  as  Ibx. 

Art.  33.  The  pmoer  of  a  quantity  is  the  product  arising  from 
multiplying  the  quantity  by  itself  one  or  more  times.  When  the 
quantity  is  taken  twice  as  a  factor,  the  product  is  called  its  square, 
or  second  power ;  when  three  times,  the  cube,  or  third  power ;  when 
four  times,  the  fourth  power,  and  so  on. 

Thus,  aX«=««,  is  the  second  power  of  a ;  ayiay^a=aaa,  is  the 
third  power  of  a ;  ayiay^ay<Ca=aaaa,  is  the  fourth  power  of  a. 

Instead  of  repeating  the  same  quantity  as  a  factor,  a  small 
figure,  called  an  exponent,  is  placed  to  the  right,  and  a  little  above 
it,  to  point  out  the  number  of  times  the  quantity  is  taken  as  a 
factor.  Thus,  aa  is  written  a^;  aaa  is  written  a^;  aaaa  is  written 
a*;  aahhh  is  written  a^h^. 

When  a  letter  has  no  expvonent,  it  is  considered  to  be  i\vQ  first,  or 
simple  power  of  the  quantity,  and  unity  is  considered  to  be  its  expo- 
nent.   Thus,  a  is  the  same  as  a\  each  expressing  the  first  power  of  a. 

Art.  34.  To  involve  or  raise  a  quantity  to  any  given  power,  is 
to  find  that  power  of  the  quantity. 

Art.  35.  The  root  of  any  quantity  is  another  quantity,  some 
power  of  which  is  equal  to  the  given  quantity.  The  root  is  called 
the  square  root,  cube  root,  fourth  root,  &c.,  according  to  the  number 
of  times  it  must  be  taken  as  a  factor  to  produce  the  given  quantity. 

Thus,  since  aX«=<^^j  therefore  a  is  the  second  root,  or  square 
root  of  a^.  In  the  same  manner,  x  is  the  third  root,  or  cube  root 
of  a;^  since  xyixy^x=--x^ . 

Art.  36.  To  extract  any  root  of  a  quantity,  is  to  find  that  root. 

Art.  37.  The  sign  ]/,  is  called  the  radical  sign.  When  placed 
before  a  quantity  it  indicates  that  its  root  is  to  be  extracted. 

Thus  ija,  or  -/a,  denotes  the  square  root  of  a;  %/a,  denotes 
the  cube  root  of  a  ;   ^a  denotes  the  fourth  root  of  a. 

Art.  38.  The  number  placed  over  the  radical  sign  is  called  the 
index  of  the  root.     Thus,  2  is  the  index  of  the  square  root,  3  of  the 

Review. — 32.  When  a  quantity  has  no  coeflScient  written,  what  coef- 
ficient is  understood  ?  33.  What  is  the  power  of  a  quantity  ?  What  is  meant 
by  the  second  power  of  a?  By  the  third  power  of  o?  What  is  an  expo- 
nent ?  For  what  is  it  used  ?  How  many  times  is  x  taken  as  a  factor  in  x^? 
Inx'?  In  x*?  Where  no  exponent  is  written,  Avhat  exponent  is  under- 
stood? 35.  What  is  the  root  of  a  quantity?  37.  What  is  the  sign  y/ 
called,  and  what  does  it  denote  ? 


30  RAY'S   ALGEBRA,    PART   FIRST. 

cube  root,  4  of  the  fourth  root,  and  so  on.  When  the  radical  has 
no  index  over  it,  2  is  understood. 

Thus,  i/9=3,  V^^^S,  V  16^:2. 

Art.  39<»  Every  quantity  written  in  algebraic  language,  that  is, 
by  means  of  algebraic  symbols,  is  called  an  algebraic  qiianiiiy,  or 
an  algebraic  expression.     Thus, 

3a  is  the  algebraic  expression  of  3  times  the  number  a ;  3a — 46, 
is  the  algebraic  expression  for  3  times  the  number  a,  diminished 
by  4  times  the  number  6 ;  2a^+3a6,  is  the  algebraic  expression  for 
twice  the  square  of  a,  increased  by  3  times  the  product  of  the 
number  a  by  the  number  b. 

Art.  40.  An  algebraic  quantity  not  united  to  any  other  by  the  sign 
of  addition  or  subtraction,  is  called  a  monomial,  or  a  quantity  of  one 
term,  or  simply  a  term.  A  monomial  is  sometimes  called  a  simple 
quaniitij.  Thus,  a,  3a,  — d^b,  '2any^,  are  monomials,  or  simple 
quantities. 

Art.  41.  An  algebraic  expression  composed  of  two  or  more 
terms,  is  called  a  polynomial,  or  a  compound  quantity. 

Thus,  c-\-2d — b  is  a  polynomial. 

Art.  42.  A  polynomial  composed  of  two  terms,  is  called  a 
binomial.     Thus,  a-\-b,  a — b,  and  c^ — d  are  binomials. 

A  binomial,  in  which  the  second  term  is  negative,  as  a — b,  is 
sometimes  called  a  residual  quantity. 

Art.  43.  A  polynomial  consisting  of  three  terms,  is  called  a 
trinomial.     Thus,  a-{-b-\-c,  and  a — b — c  are  trinomials. 

Art.  44.  The  numerical  value  of  an  algebraic  expression  is  the 
number  obtained,  by  giving  particular  values  to  the  letters,  and 
then  performing  the  operations  indicated. 

In  the  algebi-aic  expression  2a+36,  if  a=4,  and  6=5,  then 
2tt=8,  and  36=15,  and  the  numerical  value  is  8+15=23. 

Art.  45.  The  value  of  a  polynomial  is  not  aiSfected  by  chang- 
ing the  order  of  the  terms,  provided  each  term  retains  its  respec- 
tive sign.  Thus,  a"'+2a+6  is  the  same  as  b-\-d^-\-2a.  This  is 
self-evident. 

Art.  46.  Each  of  the  literal  factors  of  any  simple  qviantity  or 
term  is  called  a  dimension  of  that  term  ;  and  the  degree  of  any 
term  depends  on  the  number  of  its  literal  factors. 

Thus,  ax  consists  of  two  literal  factors,  a  and  x,  and  is  of  the 
second  degree.     The  quantity  a'^6  contains  three  literal  factors,  a,  a, 

R  E  v  I  E  w, — 38.  What  is  the  number  place<l  over  the  radical  sign  called  ? 
.39.  What  is  an  algebraic  quantity?  40.  What  is  a  monomial?  What  is  a 
simplo  quantity  ?  41.  What  is  a  polynomial  ?  42.  A  binomial?  A  residual 
quantity?  4o.  A  trinomial?  44.  What  is  meant  by  the  numerical  value 
of  an  algebraic  expre.^sion? 


DEFINITIONS    AND    NOTATION.  31 

and  b,  and  is  of  the  third  degree.     2a^x'^  contains  five  literal  factors, 
a,  a,  a,  x,  and  x,  and  is  of  the  ffth  degree;  and  so  on. 

Art.  47.  A  polynomial  is  said  to  be  homogeneous,  when  each 
of  its  terms  is  of  the  same  degree. 

Thus,  the  quantity  2a— 36-f  c  is  of  the  first  degree,  and  homo- 
geneous ;  a^+36c4-a;?/,  is  of  the  second  degree,  and  homogeneous ; 
3? — 8ay^  is  of  the  third  degree,  and  homogeneous ;  a^-\-x^  is  not 
homogeneous. 

Art.  48.  A  pai^enthesis,  (  ),  is  used  to  show  that  all  the  terms 
of  a  compound  quantity  are  to  be  considered  together  as  a  single 
term. 

Thus,  4 (a — h)  means  that  a — b  is  to  be  multiplied  by  4 ;  (a+x) 
{a — a;)  means  that  a-j-x  is  to  be  multiplied  by  a — x;  10 — {a-\-c) 
means  that  a-\-c  is  to  be  subtracted  from  10;  {a—b)'^  means  that 
a — b  is  to  be  raised  to  the  second  power ;  and  so  on. 

Art.  49.  A  vinculum, ,  is  sometimes  used  instead  of  a 

parenthesis.     Thus,  a — by(,x  means  the  same  as  {a — b)x.     Some- 
times the  vinculum  is  placed  vertically,  it  is  then  called  a  bar. 

Thus,     a  ]f-  hits  the  same  meaning  as  (a — .r+4)?/^ 
— x 
+4 

Art.  50.  Similar,  or  like  quantities  are  those  composed  of  the 
same  letters,  affected  with  the  same  exponents.  Thus,  7ab  and 
— Sab,  also  4a^6'^  and  7a^b^,  are  similar  terms.  The  quantities 
2a^b  and  2ab'^  are  not  similar,  for,  though  they  are  composed  of 
the  same  letters,  yet  these  letters  have  different  exponents. 

Art.  51.  The  reciprocal  of  a  quantit}^  is  unity  divided  by  that 

quantity.     Thus,  the  reciprocal  of  2  is  ^,  and  of  a  is  -. 

Art.  52.  The  same  letter  accented,  is  often  used  to  denote 
quantities  which  occupy  similar  positions  in  different  equations  or 
investigations.  Thus,  a,  a',  a",  a'",  represent  four  different  quan- 
tities ;  of  which  a'  is  read,  a  prime ;  a"  is  read,  a  second ;  a'"  is 
read,  a  third,  and  so  on. 

EXAMPLES. 

The  following  examples  are  intended  to  exercise  the  learner  in 
the  use  and  meaning  of  the  signs. 

Review. — 46.  What  is  the  dimension  ef  a  term?  On  what  does  the 
degree  of  a  term  depend?  What  is  the  degree  of  the  term  x;/?  Ofxi/z? 
0£2axy?  47.  When  is  a  polynomial  said  to  be  homogeneous?  48.  For 
what  is  a  parenthesis  used?  49.  What  is  a  vinculum,  and  for  what  is  it 
used?  50.  What  are  similar,  or  like  quantities  ?  51.  What  is  the  recipro- 
cal of  a  quantity  ?  52.  When  a  letter,  as  a',  has  one  accent,  what  does  it 
represent,  and  how  is  it  read?     How  is  o  with  two  acconts  read? 


32 


RAY'S   ALGEBRA,    PART   FIRST. 


Let  the  pupil  copy  each  example  on  his  slate,  or  on  the  black- 
board, and  then  express  it  in  common  language. 

Also,  let  the  numerical  value  of  each  expression  be  found,  on 


the  supposition  that  a=4,  &=3 

1.  c+d—l 

2.  4a— X. 

3.  — Sax. 

4.  Ga'x. 
a-{-c 


5. 


d=10,  x=2,  and  y 


.  Ans.  12. 
.  Ans.  14. 

6. 

h-^c+d- 
a 

Ans.  —24. 
Ans.  192. 

7. 

ay    cd 
b'^x' 

.    Ans.  3. 

8. 
9. 

3a'+2cx- 
a{a+b). 

=6. 

Ans.  4. 

Ans.  33. 


Ans 
Ans. 


41. 

28. 


10.  a+bXa—b Ans.  13. 

11.  {a^b){a—b) Ans.  7. 

12.  x''—S{a+x){a—x)+2by Ans.  4. 

2  (a — a;' 


13.  3ax- 


14. 


2ax' 


S{a+x) 


—4i/2ax. 


Ans.  7|. 


-Qx\/a. 


Ans.  —16. 


[a—xf 

In  case  further  exercises  should  be  required  to  teach  the  pupils 
the  use  of  the  signs,  the  following  equivalent  expressions  may  be 
employed,  in  which  each  letter  may  have  any  value  whatever, 
provided  that  the  same  value  be  attributed  to  the  same  letter 
throughout  the  same  expression. 

15.  3(a+c)(a— c)=3a2— 3c2. 


16.  5(a— 6)2=5a2— 10a6+56^ 

/7.2 T^ 

17. 

18. 


a+x 


^-x^y+xy^^if. 


Examples  in  which  words  are  to  be  converted   into  algebraic 
symbols. 

1.  Three  times  a,  plus  b,  minus  four  times  c.     Or,  three  into  a, 
plus  b,  minus  4  into  c. 

2.  Five  times  a,  divided  by  three  times  b. 

3.  a  minus  b,  into  three  times  c. 

4.  a,  minus  three  times  b  into  c. 

5.  a  plus  b,  divided  by  three  c. 

6.  a,  plus  b  divided  by  three  c. 

7.  5  into  a  minus  three  into  b,  divided  by  c  minus  d. 

8.  a  squared,  minus  three  a  into  b,  plus  5  times  c  into  d  squared, 

9.  X  cubed  minus  b  cubed,  divided  by  x  squared  minus  b  squared. 

10.  Five  a  squared,  into  a  plus  6,  into  c  minus  d,  minus  three 
times  X  fourth  power. 

1 1 .  a  fifth  power  minus  b  fifth  power,  divided  by  a  minus  6, 
raised  to  the  fifth  power. 


EXAMPLES  IN   NOTATION.  33 


12.  a  squared  plus  b  squared,  divided  by  a  plus  6,  squared. 

13.  The  square  root  of  a,  minus  the  square  root  of  x. 

14.  The  square  root  of  a,  minus  x. 

15.  The  square  root  of  a  minus  x. 

16.  The  square  root  of  b  squared  minus  four  into  a  into  c. 


ANSWERS. 


1. 

3a+6-4c. 

9 

5a 

36' 

3. 

{a—b)Sc. 

4. 

a—Sbc. 

5. 

a+b 
3c  • 

6. 

<■ 

7 

5a— Sb 

c—d  ' 

8. 

a:'—Sab-{-5cdK 

x^~b'' 
10.  5a'{a+b){c—d)—Sx\ 

{a-bf 

12    ^±^ 

[a-i-bf 

13.  i/a—^x. 

14.  i/a — X. 

15.  /(a-x). 

16.  i/(6-^— 4ac). 


ADDITION. 

Art.  53.  Addition  in  Algebra,  is  the  process  of  collecting  two 
or  more  algebraic  quantities  into  one  expression,  called  their  sum. 

CASE  I. 

When  the  qiianUties  are  similar,  arid  have  the  same  sign. 

1.  James  has  3  pockets,  each  containing  apples ;  in  the  first  he 
has  3  apples,  in  the  second  4  apples,  and  in  third  5  apples. 

In  order  to  find  how  many  apples  he  has,  suppose  he  proceeds  to 
find  their  sum  in  the  following  manner :     3  apples, 

4  apples, 

5  apples, 
12  apples. 

Suppose,  however,  that,  instead  of  writing  the  word  apples,  ho 
should  merely  use  the  letter  a,  thus :     3a 

4a 

5a 

12a 

Review. — 53.  What  is  algebraic  addition ?  When  quantities  are  simi' 
lar,  and  have  the  same  sign,  how  are  they  added  together  ?     • 


34  RAY'S   ALGEBRA,    PART   FIRST. 

It  is  evident  that  the  sum  of  3  times  a,  4  times  a,  and  5  times 
a,  would  be  12  times  a,  or  12a,  whatever  a  might  represent. 

2.  In  the  same  manner  the  sum  of  — 3a,  — 4a,  and  — 5a  — 3a 
would  be  — 12a.  —4a 

—5a 


Hence,  the  — 12a 

RULE, 

FOR    ADDING    SIMILAR   QUANTITIES    WITH    LIKE    SIGNS. 

Add  together  the  coefficients  of  the  several  quantities,  and  to  their 
sum  annex  the  common  letter,  or  letters,  prefixing  the  common  sign. 

Note  1. — Let  the  pupil  be  reminded,  that  when  a  quantity  has  no  coef- 
ficient prefixed,  1  is  understood;  thus,  a  is  the  same  as  la. 

Note  2. — Let  the  pupil  also  be  reminded,  that  the  sum  of  any  number 
of  quantities  is  the  same,  in  whatever  order  they  are  taken.  This  is  self- 
evident;  but  it  maybe  illustrated  by  numbers  in  the  following  manner. 
Suppose  it  is  required  to  find  the  sum  of  the  numbers  16,  25,  and  31;  in 
adding  these  numbers  together,  they  may  be  written  in  six  different  ways, 
in  each  of  which  the  sum  is  the  same.  ■  Thus: 

10  16  25  25  34  34 

25  34  IG  34  16  25 

34  25  3£  IjG  25  \Q^ 

75  75~  75  75  75  75 


EXAM 

PLES. 

3. 

4. 

5. 

6. 

3a 

—Qxy 

2a-^ 

—Sa'b 

2a 

—XIJ 

3a'^ 

—4a'b 

a 

—4xy 

5a2 

— 5a^6 

ha 

-3ry 

7a' 

— 2a7> 

^Ua 

-14xy 

lla'- 

-Ua'b 

Sum 

In  the  first  example,  Ave  will  suppose  a=2,  then  3a=3X2=6, 
2a=2x2=4,a=2,5a=5X2=10;  their  sum  is  6+4+2+ 10=:22. 

But  the  sum,  22,  is  more  easily  found  from  the  algebraic  sum, 
11a,  for  lla--llX2-=22. 

In  the  second  example,  let  a:=^3  and  y=2,  and  the  value  of  its 
terms  will  be  6x?/-=6X3X2=^36 

Xljr=:        3X2-:    6 

4x//-:4X3X2--24 

3xi/=3XSx2j=lS 

the  sum  of  their  values  is     =84 

But  this  sum  is  more  easily  found  from  the  algebraic  sum ;  for 

Revtkw. — When  several  quantities  are  to  bo  added  together,  is  tha 
result  aticc-tcd  by  the  onlor  in  wliich  they  arc  taken? 


ADDITION.  35 


14x^=14X3X2=84.     As  all  these  terms  are  negative,  their  sum 
is  -84. 

In  the  fifth  example  let  a  represent  3  feet,  then 
2a'^=2aa=2X3X3=18  square  feet, 
3a2=3aa=3X3X3=27      " 
5a2=5aa=5X3X3=45      " 
7a-=7aa=7X3X3=63      " 
and  their  sum  is  153      "         " 

Or  the  sum  =17a'=17X3X3=153  square  feet. 

Note. — It  is  recommencled  to  the  learner,  thus  to  exemplify  the  exam- 
ples numerically,  by  assigning  certain  values  to  the  letters ;  observing 
throughout  each  example,  to  adhere  to  the  same  numerical  value  for  tho 
same  letter. 

What  is  the  sum 

7.  Of  3&,  56,  76,  and  96?    Ans.  246. 

8.  Of  2a6,  5a6,  8a6,  and  lla6?     Ans.  26a6.      . 

9.  Of  ahc,  Sabc,  lobe,  and  12a6c?     Ans.  2Sabc. 

10.  Of  5a  dollars,  8a  dollars,  11a  dollars,  and  13a  dollars? 

Ans.  37a  dollars. 

11.  Of  — 3aa;,  — 5ax,  — 7aa;,  and  — 4aa:?     Ans.  — 19aa;. 

12.  Of  —bij,  —26?/,  —56?/,  and  —863/?     Ans.  — 166y. 

13.  14.  15. 

3.:7-f7  8x— 4?/  Sa'-2ax 

ay+8  5a; — 3?/  5a-— 3ax- 

2a?/+4  Ix—Gy  7a2-5ax 

5a?/+6  6.T — 2?/  4a"^ — 4ax 

CASE   II. 

Art.  54.    When  quantities  are  alike,  bid  have  unlike  signs. 

1.  If  Jame;^  receives  from  one  man  6  cents,  from  another  9 
cents,  and  from  a  third  10  cents;  and  then  spends,  for  candy  4 
cents,  and  for  apples  3  cents,  how  much  money  will  he  have  left? 

If  the  quantities  he  received  be  considered  positive,  then  those 
he  spent  may  ho  considered  negative ;  and  the  question  is,  to  find 
the  sum  of  +6^;,  -{-{h,  -flOV-,  — 4c  and  — 3<",  which  may  be  written 
thus:      +6c 

+9c        Hero,  it  is  evident,  the  true  result  will  be  found,  by 

4-lOc    adding  the  positive  quantities  into  one  sum,  and  the 

— 4c    negative  quantities  into  another,  and  then  taking 

—3c    their  difference.     It  is  thus  found  that  he  received 

Z^lSc    25c,  and  spent  7c,  which  left  18c. 

2.  Suppose  James  should  receive  5  cents,  and  then  spend  7 
cents,  what  sum  would  he  have  left? 


36  KAY'S    ALGEBRA,    PART    FIRST. 

If  we  denote  the  5c  as  positive,  the  7c  will  be  negative,  and  it  is 
required  to  find  the  sum  of  +5c  and  — 7c. 

In  its  present  form,  however,  it  is  evident  that  the  question  is 
impossible.  But  if  we  suppose  that  James  had  a  certain  sum  of 
money  before  he  received  the  5c,  we  may  inquire  how  much  less 
money  he  had  after  the  operation,  than  before  it;  or,  in  other  words, 
what  effect  the  operation  had  upon  his  money.  The  answer,  it  is 
obvious,  would  be,  that  his  money  was  diminished  2  cents ;  this 
would  be  indicated  by  the  sum  of  +5c  and  — 7c,  being  — 2c. 

It  is  thus  we  say,  that  the  sum  of  a  positive  and  negative  quan- 
tity is  equal  to  the  difference  between  the  two  ;  the  object  being  to 
find  what  the  united  effect  of  the  two  Avill  be  upon  some  third  quan- 
tity.    This  may  be  further  illustrated  by  the  following  example. 

3.  A  merchant  has  a  certain  capital ;  during  the  year  it  is  hv- 
creased  by  3a  and  8a  dollars,  and  diminished  by  2a  and  5a  dollars ; 
how  much  will  his  capital  be  increased  or  diminished  at  the  close 
of  the  year  ?  ' 

If  we  denote  the  gains  as  positive,  the  losses  will  be  negative. 
The  sum  of  +3a,  4-8«,  —2a,  and  —5a  is  11a— 7a,  which  is  equal 
to  +4a.  Hence,  we  say,  that  the  merchant's  capital  will  be  in- 
creased by  4a  dollars ;  and  whatever  the  capital  may  have  been, 
the  result  will  be  the  same  to  increase  it  by  4a  dollars,  as  first  to 
increase  it  by  3a  and  8a  dollars,  and  then  to  diminish  it  by  2a  and 
5a  dollars.  Had  the  loss  been  greater  than  the  gain,  the  efiect 
would  bo  to  diminish  the  capital ;  and  this  would  be  indicated,  by 
the  sum  of  the  gains  and  losses  being  negative. 

If  the  gain  and  loss  were  equal,  it  is  evident  the  capital  would 
neither  be  increased  nor  diminished ;  or,  in  other  words,  if  the 
amount  of  the  positive  quantities  was  equal  to  that  of  the  negative, 
their  sum  would  be  0.  Thus,  +3a— 3a=0.  If  a==4,  +3a=+12 
and  — 3a=— 12,  and  +12—12=0. 

From  this  the  pupil  will  perceive,  that  to  add  a  negative  quan- 
tity is  the  same  as  to  subtract  a  positive  quantity.  In  such  cases, 
the  process  of  addition  is  called  algebraic  addition,  and  the  sum  is 
called  the  algebraic  sum,  to  distinguish  them  from  arithmetical 
addition,  and  arithmetical  sum.     Hence,  the 

RULE, 

FOR  THE  ADDITION  OF  QUANTITIES  WHICH  ARE  ALIKE,  BUT  HAVE  UNLIKE  SIGNS. 

Find  the  sum  of  the  coefficients  of  the  similar  positive  quantities  ; 
also,  the  sum  of  the  coefficients  of  the  similar  negative  quantities. 
Subtract  the  less  sum  from  the  greater  ;  then,  to  the  difference  prefix 
the  sign  of  the  greater,  and  annex  the  common  literal  part. 


ADDITION.  37 


4.  "What  is  the  sum  of  -\-Sa,  —5a,  +9a,  —6a,  and  +7a? 
Here,  the  sum  of  the  coefficients  of  the  positive  terms,  is 

3+9+7=-+19 
The  sum  of  the  coefficients  of  the  negative  terms,  is 

-5-6=-ll 
The  difference  between  19  and  11  is  8,  to  which,  prefixing  the 
sign  of  the  greater,  and  annexing  the  literal  part,  we  have  for  the 
required  sum  -\-Sa. 

In  practice,  it  is  most  convenient  to  write  the      3a 
different  terms  under  each  other.     Thus,  — 5a 

9a 
6a 
7a 
Sum=8a 
Beginners,  however,  will  sometimes  find  it  easier      Sa — 5a 
to  arrange  the  positive  quantities  in  one  column,       9a — 6a 
and  the  negative  in  another.     The  preceding  ex-      7a 
ample  may  be  arranged  as  in  the  margin.  j^g^ lla=^8a 

EXAMPLES. 

5.  What  is  the  sum  of  8a  and  — 5a  ?     Ans.  3a. 

6.  What  is  the  sum  of  5a  and  — 8a  ?     Ans.  — 3a. 

7.  What  is  the  sum  of  — 7ax,  Sax,  6ax,  and  — ax  ?      Ans.  ax. 

8.  What  is  the  sum  of  5abx,  — 7a6a:,  3a5x,  — abx,  and  4abx1 

Ans.  4abx. 

9.  Add  together,  4ac,  5ac,  — 3ac,  7ac,  —Gac,  — 2ac,  9ac,  and 
—  17ac.  Ans.  — Sac. 

10.  Find  the  sum  of  6a— 46,  3a+26,  —7a—Sb,  and  — a+06. 

Ans.  a — /). 

11.  Find  the  sum  of  Sax—2by,  —2ax-^Sby,  Sax—4tbij,  and 
—9ax-\-Sby.  Ans.  5hy. 

12.  Find  the  sum  of  Sab—\Ox,  -Sab+lx,  Sab—6x,  —ab+2x, 
and  — 2a6+7x.  Ans.  0. 

13.  Find  the  sum  of  4a'—2b,  —6a'-\-2b,  2a'—Sb,  —5a'—Sb, 
and  — 3a2+96.  Ans.  —Sa'—2b. 

14.  Find  the  sum  of  xy—ac,  Sxy—9ac,  —7xy+5ac,  4xy+6ac, 
and  —xy—2ac.  Ans.  —ac 

Note. — The  operation  of  collecting  the  similar  terms  in  any  algebraic 
expression  into  one  sum,  as  exemplified  in  this  case,  is  sometimes  called 
the  deduction  of  Polynomiala.     The  following  are  examples. 

15.  Reduce  3a64-5c— 7a6+8c+8a6— 14c— 2a6+c  to  its  sim- 
plest form.  Ans.  2ab. 

Review. — 54.  How  are  quantities  added  together  that  are  similar,  hut 
have  unlike  signs  ? 


38  RAY'S   ALGEBRA,    PART    FIRST. 


16.  Reduce  5a'c—Sb^  +  4a'c+5b^—8a^c  +  2b^  to  its  simplest 
form.  Ans.  a^c+46^ 

CASE  III. 

Art.  55.  WJien  the  quantities  are  unlike,  or  partly  like  and 
partly  unlike. 

1.  Thomas  has  a  marbles  in  one  hand,  and  h  marbles  in  the 
other ;  what  expression  will  represent  the  number  in  both  ?  If  a 
is  represented  by  3,  and  h  by  4,  then  the  number  in  both  would  be 
represented  by  3-1-4,  or  7. 

In  the  same  manner,  the  number  in  both  would  be  represented 
by  a-^h ;  but  unless  the  numerical  values  of  a  and  b  are  given,  it 
is  evidently  impossible  to  represent  their  sum  more  concisely,  than 
by  a+6. 

In  the  same  manner,  the  sum  of  the  quantities  a-{-b  and  c-[-d,  is 
represented  by  a^b-\-c-{-d. 

If,  in  any  expression,  there  are  two  or  more  like  quantities,  it  is 
obvious,  that  they  may  be  reduced  to  a  single  expression  by  the 
preceding  rules.  Thus,  the  sum  of  2a4-x  and  ^a-\-y,  is  equal  to 
2a-|-3a-|-a;-hy,  which  reduces  to  ^a-\-x-\-y. 

It  is  evident  that  this  case  embraces  the  two  preceding  cases ; 
hence,  the  "• 

GENERAL   RULE. 
FOR    THE    ADDITION    OF    ALGEBRAIC    QUANTITIES. 

WHte  the  quantities  to  be  added,  placing  those  that  are  similar 
under  each  other  ;  then^reduce  the  similar  terms,  and  annex  the  other 
terms  with  their  proper  signs. 

Remark.  — If  a  reason  is  asked  for  placing  similar  terms  under  each 
other,  the  reply  is,  that  it  is  not  absolutely  necessary ;  but  as  we  can  only  add 
similar  terms  together,  it  is  a  matte)-  of  convenience,  to  place  them  under 
each  other. 

EXAMPLES. 

Add  together 

2.  6a— 4c-f  36,  and  —2a— 3c— 5b.  Ans.  4a— 7c— 26. 

3.  2a64-c,  4ax— 2c+I4,  12— 2aa:,  and  Gab+Sc-x. 

Ans.  8a6+2aa;+2c-|-26— X. 

4.  Ua+x,  136—//,  — lla-f-2y,  and  — 2a-126-f-2. 

Ans.  a-\-b-\-x-{-y-{-z. 

5.  a— 6,  26— c,  2c— d,  2d-e,  and  2e+f.    Ans.  a+b+c+d+e+f. 

6.  _754_3c,  46— 2c+3x,  36— 3c,  and  2c~2.r.     Ans.  x. 

7.  3(a+6),  5(a+6),  and  7(a+6).     Ans.  15(a-l-6). 

R  E  v  1  E  w. — 55.  What  is  the  general  rule  for  the  addition  of  algebraic  quan- 
tities?    In  writing  them,  why  are  similar  quantities  placed  under  each  other? 


ADDITION.  39 

Note. — The  learner  should  be  reminded,  that  the  quantities  in  the 
parentheses  are  to  be  considered  as  one  quantity ;  then  it  is  evident,  that 
3  times,  5  times,  and  7  times  any  quantity  whatever,  will  bo  equal  to  15 
tinges  that  quantity. 

Add  together 

8.  3a(6+x),  5a(&+x),  la[b-^x),  and  -na{b-\-x). 

Ans.  Aa{h-\-x). 

9.  2c(a2-52),  _3c(a^_&2),  i^c{d'-h''),  and  —Ac[a:'~W). 

Ans.  c{o? — ¥). 

10.  3«z— 46?/— 8,  — 2a2;+56y+6,  baz-\-Qhy—l,  and  —Saz 
—7b^+b.  Ans.  —2az-4. 

11.  Sax—Scz\  —5ax-\-5cz^,  ax-\-2cz^,  and  —4ax—4cz'^.     Ans.  0. 

12.  8a+6,  2a— 6+c,  — 3a-f-56+2cZ,  -66— 3c+3c?,  and  —5a 
+7c—2d.  Ans.  2a— 6+5c+3^. 

13.  7x-6y+52+3-^, -x-Sf/S-g—x+y—Sz-l -{-Ig,  —2x 
-r-3y+3z— 1  — ^r,  and  a:+8y— 5^+9+ (/.  Ans.  4x+3?/+2+5^. 

14.  2a}-Y^ab—xy,  — 7a^+3a6— 3xy,  — 3a2— 7a6+5xy,  and  9a''' 
— db—2xy.  Ans,  o?—xy. 

15.  5aW— 8a26'  +  a:V+a-7/2,  4a26'— 7a362_3ay -f  Gx^,  Sa^^^ 
+3a263— 3a;V+5x2/^  and  2d'b^—a?¥—2x'y—Zxy^.    Ans.  a'V'+x'y. 


SUBTRACTION. 


Art.  56.  Subtraction  in  Algebra,  is  the  process  of  finding  the 
simplest  expression  for  the  difference  between  two  algebraic 
quantities. 

In  Algebra,  as  in  Arithmetic,  the  quantity  to  be  subtracted  is 
called  the  subtrahend.  The  quantity  from  which  the  subtraction 
is  to  be  made,  is  called  the  minuend.  The  quantity  left,  after  the 
subtraction  is  performed,  is  called  the  difference,  or  remainder. 

Remark. — The  word  subtrahend  means,  to  he  subtracted;  the  word 
minuend,  to  be  diminished. 

1.  Thomas  has  5a  cents ;  if  he  give  2a  cents  to  his  brother, 
how  many  will  he  have  left? 

Since  5  times  any  quantity,  diminished  by  2  times  the  same 
quantity,  leaves  3  times  the  quantity,  the  answer  is  evidently  3a; 
that  is  5a — 2a=3a. 

Hence,  to  find  the  difference  between  two  positive  similar  quan- 
tities, wejind  the  difference  between  their  coefficients,  and  prefix  it  to 
the  common  letter,  or  letters. 


40  RAY'S    ALGEBRA,    PART   FIRST. 

Let  it  be  noted,  that  the  sign  of  the  quantity  to  be  subtracted, 
is  changed  from  plus  to  minus. 

2.  3.  4.  5. 

From  5ic  lab  Sxy  Wa^x  ' 

Take  3a;  Sab  bxy  fia?x 

Remainder  2a;  Aab  Sxy  Qd^x 

6.  From  9g5,  take  4a Ans.  5a. 

7.  From  116,  take  116 Ans.  0. 

8.  From  Waxy,  take  ^axy Ans.  Saxy. 

9.  From  \2bcx,  take  bbcx Ans.  Ibex. 

10.  From  137imj7,  take  Qy^wj? Ans.  4A7np. 

11.  From  3a^  take  2a^ Ans.  a^ 

12.  From  76%,  ta,ke  462a;y .    .  Ans.  36%. 

Art.  57. — 1.  Thomas  has  a  number  of  apples,  represented  by 

a ;  if  he  give  away  a  quantity,  represented  by  6,  what  expression 
will  represent  the  number  of  apples  he  has  left  ? 

If  a  represents  6,  and  6  4,  then  the  number  left  would  be  repre- 
sented by  6 — 4,  which  is  equal  to  2 ;  and  whatever  numbers  a 
and  6  represent,  it  is  evident  that  their  difference  may  be  ex- 
pressed in  the  same  w^ay,  that  is,  by  a — 6. 

Hence,  to  find  the  difference  between  two  quantities  that  are  not 
similar,  ive  place  the  sign  minus  before  the  quantity  that  is  to  be 
subtracted. 

Let  the  pupil  here  notice  again,  that  the  sign  of  the  quantity  to 
be  subtracted,  is  changed  from  plus  to  minus. 

2.  From  c,  take  d Ans.  c — d. 

3.  From  2m,  take  3?i Ans.  2m — 3?i. 

4.  From  56,  take  3c Ans.  56— 3a 

5.  From  a6,  take  cd Ans.  ab — cd. 

6.  From  a^x,  take  ax^ Ans.  d'x — ax^. 

7.  From  x^,  take  x Ans.  x^ — x. 

8.  From  xy,  take  yz Ans.  xy — yz. 

Art.  58.^1.  Let  it  be  required  to  subtract  5+3  from  9. 

If  we  subtract  5  from  9,  the  remainder  will  be  9 — 5  ;  but  we 
wish  to  subtract,  not  only  5,  but  also  3  ;  hence,  after  we  have  sub- 
tracted 5,  we  must  also  subtract  3  ;  this  gives  for  the  remainder, 
9—5 — 3,  which  is  equal  to  1. 

Review. — 56.  What  is  Subtraction  in  Algebra?  What  is  the  quantity 
to  be  subtracted,  called  ?  What  is  the  quantity  called,  from  which  the  sub- 
traction is  to  be  made  ?  What  does  subtrahend  mean  ?  What  does  minu- 
end mean  ?  How  do  you  find  the  difference  between  two  positive  similar 
quantities  ?  57.  How  do  you  find  the  difi"erenco  between  two  quantities 
that  are  not  similar  ? 


SUBTRACTION.  41 


2.  Again,  suppose  that  it  is  required  to  subtract  5 — 3  from  9 
If  we  subtract  5  from  9,  the  remainder  will  be  9 — 5 ;  but  the 
quantity  to  be  subtracted  is  3  less  than  5,  and  we  have,  therefore, 
subtracted  3  too  much ;  we  must,  therefore,  add  3  to  9 — 5,  which 
gives  for  the  true  remainder,  9 — 5+3,  which  is  equal  to  7. 

3.  Let  it  now  be  required  to  subtract  h — c  from  a. 

If  we  take  h  from  a,  the  remainder  is  a — h ;  but,  in  doing  this, 
we  have  subtracted  c  too  much ;  hence,  to  obtain  the  true  result, 
we  must  add  c.     This  gives,  for  the  true  remainder,  a — 6+c. 

If  a=9,  6^=5,  and  c=3,  the  operation  and  illustration  by  figures 
Avould  stand  thus :    from  a  from  9  =9 

take  6 — c  take  5 — 3  =2 

Remainder,  a — 6+c        Rem.     9 — 5+3  =7 

The  same  principle  may  be  further  illustrated  by  the  following 
examples. 

4.  a—(^ — a)  =a—c-\-a  =2a — c. 
a — [a — c)  =a — a+c  =c. 
a+b—{a—b)  =a+h—a-\-b  =2b. 

Let  it  be  noted,  that  in  the  result  in  each  of  the  preceding  ex- 
amples, the  signs  of  the  quantity  to  be  subtracted  have  been 
changed  from  plus  to  minus,  and  from  minus  to  plus ;  hence,  in 
order  to  subtract  a  quantity,  it  is  merely  necessary  to  change  the 
signs  and  add  it.    Hence,  the 

RULE, 
FOR    FINDING    THE    DIFFERENCE  BETWEEN  TWO  ALGEBRAIC  QUANTITIES. 

Write  the  quantity  to  be  subtracted  under  thai  from  which  it  is  to 
be  taken,  placing  similar  terms  under  each  other.  Conceive  the  signs 
of  all  the  terms  of  the  subtrahend  to  be  changed,  and  then  reduce  the 
result  to  its  simplest  form. 

Note. — It  is  a  good  plan  with  beginners,  to  direct  them  to  write  the 
example  a  second  time,  and  then  actually  change  the  signs,  and  add,  as  in 
the  following  example.  They  should  do  this,  however,  only  till  they  become 
familiar  with  the  rule. 

From  5a+36 — c  The  same,  with  the         5«+36— c 

Take  2a— 2/>— 3c     signs   of  the   subtra-  — 2rt+2/)+3c 
llemain.     3a+56+2c     hend  changed.  3a+56+2c 

EXAMPLES. 

6.  7.  8. 

From  3ax— 2?/  Acx'—'^by^  8xyz+3a2— 8 

Take  2aa:+3.y  2cx  —Sb^f  ryxgz—Saz+8 

Remainder,        ax— by  4.cx^—2cx  ^xyz-^iSaz—lQ 
4 


42  RAY'S   ALGEBRA,    PART    FIRST. 

9.  10.  11. 

rrom7a:+4y  3a- -26  Gax-4f-^3 

Take  6x — y  5a  -  -36  3aa: — 6//^-f2 

12.  From  14,  take  ah  -f> Ans.  19 — ah. 

13.  From  a^h,  take  a Ans.  h. 

14.  From  a,  take  a^h Ans.  — 6. 

15.  From  X,  take  x — 5 Ans.  5. 

16.  From  3ax*,  take  2ax+7 Ans.  ax — 7. 

17.  From  x-\-y,  take  x — y Ans.  2//. 

18.  From  x — y,  take  x-\-y Ans.  — 2?/. 

19.  From  x — y,  take  y — x Ans.  2x— 2y. 

20.  From  x-^y-\-z,  take  x — y — z Ans.  2y-\-2z. 

21.  From  5x+3y — z,  take  4x-\-3y-\-z Ans.  x — 2z. 

22.  From  a,  take  — a Ans.  2a. 

23.  From  8a,  take  — 3a Ans.  11a. 

24.  From  a,  take  — 4a Ans.  5a. 

25.  From  56,  take  116 Ans. —66. 

26.  From  a,  take  — 6 Ans.  a+6. 

27.  From  3a,  take  —26 Ans.  3a+26. 

28.  From  —9a,  take  3a Ans.  —12a. 

29.  From  —7a,  take  —7a Ans.  0. 

30.  From  —19a,  take  —20a Ans.  a. 

31.  From —6a,  take — 5a Ans. —a. 

32.  From  —3a,  take  — 56 Ans.  — 3a+56,  or  56— 3a. 

33.  From  —13,  take  3 Ans.  —16. 

34.  From  —9,  take  —16 Ans.  7. 

35.  From  12,  take  —8 Ans.  20. 

36.  From  —14,  take  —5 Ans.  —9. 

37.  From  3a— 26+6,  take  2a— 76-3 Ans.  a-f-56+9. 

38.  From  13a— 26+9c— 3^,  take  8a-66+9c— lOJ+12. 

Ans.  5a+46+7fZ— 12. 

39.  From  — 7a+3w— 8x,  take  — 6a  — 5?/t— 2a;+3rf. 

Ans.  — a-f  8w — 6x — 3c?. 

40.  From  32a+36,  take  5a+ 176 Ans.  27a— 146. 

41.  From  6a+5— 36,  take  —2a— 96— 8.  .    .  Ans.  8a-f66+13. 

42.  From  3c— 2Z+56-,  take  8Z+7c— 4? Ans.  c— 6Z. 

43.  From  3ax— 2y^  take  — 5ax— 8?/ Ans.  8ax+6?/*. 

44.  From  2x-— 3aV-f  9,  take  x2+5aV— 3.    Ans.  x^— 8a'^xM- 12. 

45.  From  4x2^— 5c2;+8m,  take  —cz-^^xhf—Acz. 

Ans.  2  x^-f-Sm. 

46.  From  x' — llx?/z+3a,  take  —Qxyz-\-l — 2a — 5x^2. 

Ans.  x^+5a — 7. 

47.  From  5(x+.y),  take  2(x+y) Ans.  3(x-f  y). 


SUBTRACTION.  43 


48.  From  Sa{x — z),  take  a{x — z) Ans.  2a{x — z). 

49.  From  7a'^{c—z) — ab{c~d),  take  5a^{c—z)—5ab{c—d). 

Ans.  2a''{c—z)+4ab{c~d). 

Art.  59.  It  is  sometimes  convenient  to  indicate  the  subtraction 
of  a  polynomial  without  actually  performing  the  operation.  This 
may  be  done,  if  it  is  a  monomial,  by  placing  the  sign  minus  before 
it ;  and,  if  it  is  a  polynomial,  by  enclosing  it  in  a  parenthesis,  and 
then  placing  the  sign  minus  before  it. 

'  Thus,  to  subtract  a — b  from  2a,  we  may  write  it  2a — (a — b), 
which  reduces  to  a-j-b. 

By  this  transformation,  the  same  polynomial  may  be  written  in 
several  different  forms  ;  thus : 

a—b-^c—d=^a—b—{d — c)=a — d—{b—c)=a — {b—c-\-d). 

Let  the  pupil,  in  each  of  the  following  examples,  introduce  all 
the  quantities,  except  the  first,  into  a  parenthesis,  and  precede  it 
by  the  sign  minus,  without  altering  the  value  of  the  expression. 

1.  a—b-\-c Ans.  a—{b—c). 

2.  b+c—d Ans.  b—{d—c). 

3.  sc^ — 2xy-\-z Ans.  x^ — [2x2/ — z). 

4.  ax-{-bc — cd-\-h Ans.  ax — {cd — be — h). 

5.  w— n — z — s Ans.  m — {^n-\-z-\-s). 

6.  m — w+2+5 Ans.  m — (w — z — s). 

It  will  be  found  a  useful  exercise  for  the  pupil,  to  take  each  of 

the  preceding  polynomials,  and  without  changing  their  values, 
Avrite  them  in  all  possible  modes,  by  including  either  two  or  more 
terms  in  a  parenthesis. 


OBSERVATIONS   ON   ADDITION   AND   SUBTRACTION. 

Art.  60.  It  has  been  shown,  that  Algebraic  Addition  is  the 
process  of  collecting,  into  one,  the  quantities  contained  in  two  or 
more  expressions.  The  pupil  has  already  learned,  that  these  ex- 
pressions may  be  all  positive,  or  all  negative,  or  partly  positive  and 
partly  negative.  If  they  are  either  all  positive,  or  all  negative, 
the  sum  will  be  greater  than  either  of  the  individual  quantities  ; 
but,  if  some  of  the  quantities  are  positive  and  others  negative,  the 
aggregate  may  be  less   than  either  of  them,  or,  it  may  even  be 

Review. — In  subtractings — c  from  a,  after  taking  away  i,  have  we 
pubtraeted  too  much,  or  too  little  ?  What  must  be  added,  to  obtain  the 
true  result?  Why?  What  is  the  general  rule  for  finding  the  difference 
between  two  algebraic  quantities  ?  59.  How  can  the  subtraction  of  an 
algebraic  quantity  be  indicated  ? 


44  RAY'S   ALGEBRA,    PART   FIRST. 

nothing.     Thus,  the  sum  of  -^4a  and  — 3a  is  a ;  while  that  of 
-\-a  and  — a,  is  zero,  or  0. 

As  the  pupil  should  have  clear  views  of  the  use  and  meaning  of 
the  various  expressions  employed,  it  may  be  asked,  what  idea  is  he 
to  attach  to  the  operations  of  algebraic  addition  and  subtraction. 

Art.  61.  In  common  or  arithmetical  addition,  when  we  say, 
that  the  sum  of  5  and  3  is  8,  we  mean,  that  their  sum  is  8  greater 
than  0.  In  algebra,  when  we  say  that  5  and  — 3  are  equal  to  2, 
we  mean,  that  the  aggregate  eflfect  of  adding  5  and  subtracting  3, 
is  the  same  as  that  of  adding  2.  When  we  say,  that  the  sum  of 
— 5  and  -\-S,  is  — 2,  we  mean,  that  the  result  of  subtracting  5, 
and  adding  3,  is  the  same  as  that  of  subtracting  2.  Some  alge- 
braists say,  that  numbers  with  a  positive  sign  represent  quantities 
greater  than  0,  while  those  with  a  negative  sign,  such  as  — 3, 
represent  quantities  less  than  nothing.  The  phrase,  less  than  noth- 
ing, however,  can  not  convey  an  intelligible  idea,  with  any  signifi- 
cation that  would  be  attached  to  it  in  the  ordinary  use  of  language  ; 
but,  if  we  are  to  understand  by  it,  that  any  negative  quantity,  when 
added  to  a  positive  quantity,  will  produce  a  result  less  than  if 
nothing  had  been  added  to  it;  or,  that  a  negative  quantity,  when 
subtracted  from  a  positive  quantity,  will  produce  a  result  greater 
than  if  nothing  had  been  taken  from  it,  then  the  phrase  has  a  cor- 
rect meaning.  The  idea,  however,  would  be  properly  expressed, 
by  saying,  that  negative  quantities  are  relatively  less  than  zero. 
Thus,  if  we  take  any  number,  for  instance  10,  and  add  to  it  the 
numbers  3,  2,  1,0,  — 1,  — 2,  and  — 3,  we  see,  that  adding  a 
negative  number  produces  a  less  result  than  adding  zero. 

10  10  10  10  10  10  10 

_1      _?.      _!       _2.     ri     -?.     n^ 

13  12  11  10  9  8  7' 

From  this,  we  also  see,  that  adding  a  negative  number,  produces 
the  same  result,  as  subtracting  an  equal  positive  number. 

Again,  if  we  take  any  number,  for  example  10,  and  subtract 
from  it  the  numbers  3,  2,  1,  0,  — 1,  — 2,  and  —3,  we  see,  that 
subtracting  a  negative  number  produces  a  greater  result  than 
subtracting  zero : 

10  10  10  10  10  10  10 

_3         ^         J_  0        -1         -2        -3 

7  8  9  10  11  12  13 

From  this,  we  also  see,  that  subtracting  a  negative  number,  pro- 
duces the  same  result,  as  adding  an  equal  positive  number. 

R  E  V I E  w. — 60.  When  is  the  sum  of  two  algebraic  quantities  less  than 
either  of  them  ?     Wh»n  is  the  sum  equal  to  zero? 


OBSERVATIONS.  45 


Art.  62.  In  consequence  of  the  results  they  produce,  it  is  cus- 
tomary to  say,  of  two  negative  algebraic  quantities,  that  the  least 
is  that  which  contains  the  greatest  number  of  units.  Thus,  —3 
is  said  to  be  less  than  — 2.  But,  of  two  negative  quantities,  that 
which  contains  the  greatest  number  of  units  is  said  to  be  numeri- 
cally iha  greatest;  thus,  — 3  is  numerically  greater  than  — 2. 

Art.  63.  A  correct  idea  of  the  nature  of  the  addition  of  posi- 
tive and  negative  quantities,  may  be  gained  by  the  consideration 
of  such  questions  as  the  following: 

Suppose  the  sums  of  money  put  into  a  drawer  to  be  positive 
quantities,  and  those  taken  out  to  be  negative ;  how  will  the 
money  in  the  drawer  be  affected,  if,  in  one  day,  there  are  20  dol- 
lars taken  out,  afterwards  15  dollars  put  in,  after  this  8  dollars 
taken  out,  and  then  10  dollars  put  in?  Or,  in  other  words,  what 
is  the  sum  of — 20,  +15,  —8,  and  -flO?  The  answer,  evidently, 
is  — 3  ;  that  is,  the  result  of  the  whole  operation  diminishes  the 
amount  of  money  in  the  drawer  3  dollars.  Had  the  sum  of  the 
quantities  been  positive,  the  result  of  the  operation  would  have 
been,  an  increase  of  the  amount  of  money  in  the  drawer. 

Again,  suppose  latitude  north  of  tlie  equator  to  be  reckoned  +, 
and  that  south,  —  ;  and  that  the  degrees  over  which  a  ship  sails 
north,  are  designated  by  +,  while  those  she  sails  over  south,  are 
designated  by  — ,  and  that  we  have  the  following  question :  A 
ship,  in  latitude  10  degrees  north,  sails  5  degrees  south,  then  7 
degrees  north,  then  9  degrees  south,  and  then  3  degrees  north ; 
what  is  her  present  latitude  ? 

This  question  is  the  same  as  to  find  the  sum  of  the  quantities 
+  10,  —5,  +7,  —9,  and  +3  ;  this  is  evidently  +6;  that  is,  the 
ship  is  in  6  degrees  north  latitude.  Had  the  sum  of  the  negative 
numbers  been  tiie  greater,  it  follows,  that  the  ship  would  have 
been  found  in  south  latitude. 

Other  questions  of  a  similar  nature  may  be  used  by  the  instructor, 
to  illustrate  the  subject. 

Art.  64.  Subtraction,  in  arithmetic,  shows  the  method  of  find- 
ing the  excess  of  one  quantity  over  another  of  the  same  kind.  In 
this  case,  the  number  to  be  subtracted  must  be  less  than  that  from 
which  it  is  to  bo  taken  ;  and,  as  they  are  considered  without  refer- 

Review. — 61.  What  is  meant,  by  saying  that  the  sum  of-j-5  and  — 3, 
is  equal  to  -|-2  ?  What  is  meant,  by  saying  that  the  sum  of  — 5  and  -f-3, 
is  equal  to  — 2  ?  Is  it  correct  to  say,  that  any  quantity  is  less  than  noth- 
ing? What  is  the  eifect  of  adding  a  positive  quantity?  Of  adding  a  nega- 
tive quantity?  Of  subtracting  a  positive  quantity?  Of  subtracting  a 
negative  quantity?  62.  In  comparing  two  negative  algebraic  quantities, 
which  is  called  the  least  ?     Which  is  numerically  the  greatest  ? 


46  RAY'S    ALGEBRA,    PART    FIRST. 

ence  to  sign,  it  is  equivalent  to  regarding  them  of  the  same  sign. 
Algebraic  Subtraction  shows  the  method  of  finding  the  difference 
between  two  quantities  which  have  either  the  same  or  unlike  signs  ; 
and  it  frequently  happens,  that  this  difference  is  greater  than 
cither  of  the  quantities.  To  understand  this  properly,  requires  a 
knowledge  of  the  nature  of  positive  and  negative  quantities. 

All  quantities  are  to  be  regarded  as  positive,  unless,  for  some 
special  reason,  they  are  otherwise  doeignated.  Negative  quanti- 
ties embrace  those  that  are,  in  their  nature,  the  opposite  of  positive 
quantities. 

Thus,  if  a  merchant's  gains  are  positive,  his  losses  are  negative ; 
if  latitude  north  of  the  equator  is  reckoned  +,  that  south,  would 
be  — ;  if  distance  to  the  right  of  a  certain  line  is  reckoned  +, 
then  distance  to  the  left  would  be  —  ;  if  elevation  above  a  certain 
point,  or  plane,  is  regarded  as  -f ,  then  distance  below  would  be 
— ;  if  time  after  a  certain  hour  is  +,  then  time  before  that  hour 
is  — ;  if  motion  in  one  direction  is  +,  then  motion  in  an  opposite 
direction  would  be  — ;  and  so  on. 

With  this  knowledge  of  the  meaning  of  the  sign  minus,  it  is 
easy  to  see  how  the  difierence  of  two  quantities  having  the  same 
sign,  is  equal  to  their  difference ;  and  also,  how  the  difference  of 
two  quantities  having  different  signs,  is  equal  to  their  sum. 

1.  One  place  is  situated  10,  and  another  6  degrees  north  of  the 
equator,  what  is  their  difference  of  latitude  ? 

Here  we  are  required  to  find  the  difference  between  -f  10  5ind 
4-6,  which  is  evidently  -|-4  ;  by  which  we  are  to  understand 
that  the  first  place  is  4  degrees  farther  north  than  the  second. 

2.  Tw^o  places  are  situated,  one  in  10,  and  the  other  in  6  degrees 
south  latitude  ;  what  is  the  difference  of  latitude  ? 

Here  we  are  required  to  find  the  difference  between  — 6  and  — 10, 
which  is  evidently  — 4,  by  which  we  learn,  that  the  first  place  is 
4  degrees  farther  south  than  the  second. 

3.  One  place  is  situated  in  10  degrees  north,  and  another  in  G 
degrees  south  latitude ;  what  is  their  difference  of  latitude  ? 

Here  we  are  required  to  find  the  difference  between  -f  10  and  —  6, 
or  to  take  — 6  from  +10,  which,  by  the  rule  for  subtraction,  leaves 
-|-16;  which  is  evidently  the  difference  of  their  latitudes,  and  from 
which  we  learn,  that  the  first  place  is  16  degrees  fiirther  north 
than  the  other. 

It  is  thus,  when  properly  understood,  the  results  are  always 
capable  of  a  satisfactory  explanation. 

Review. — 64.  In  what  respects  does  algebraic  differ  from  arithmetical 
Subtraction?  In  what  respect  do  negative  quantities  differ  from  positive? 
Illustrate  the  difference  by  examples. 


MULTIPLICATION.  47 


MULTIPLICATION. 

Art.  65.  Multiplication,  in  Algebra,  is  the  process  of  taking 
one  algebraic  expression,  as  often  as  there  are  units  in  another. 

In  Algebra,  as  in  Arithmetic,  the  quantity  to  be  multiplied  ia 
called  the  imdtiplicand ;  the  quantity  by  which  we  multiply,  the 
multiplier,  and  the  result  of  the  operation,  the  product.  The  mul- 
tiplicand and  multiplier  are  generally  called  factors. 

Art.  66.  Since  the  quantity  a,  taken  once,  is  represented  by  a, 
when  taken  twice,  by  a-\-a,  or  2a,  when  taken  three  times,  by 
a-\-a-\-a,  or  3a,  it  is  evident,  that  to  midtiply  a  literal  quantity  hy 
a  number,  it  is  only  necessary  to  write  the  multiplier  as  the  coefficient 
of  the  literal  quantity. 

L  If  1  lemon  costs  a  cents,  how  many  cents  will  5  lemons  cost? 

If  one  lemon  costs  a  cents,  fve  lemons  will  cost  five  times  as 
much,  that  is  5a  cents. 

2.  If  1  orange  costs  c  cents,  how  many  cents  will  6  oranges  cost? 

3.  A  merchant  bought  a  pieces  of  cloth,  each  containing  b  yards, 
at  c  dollars  per  yard ;  how  many  dollars  did  the  whole  cost? 

In  a  pieces,  the  number  of  yards  would  be  represented  by  ab, 
or  ba,  and  the  cost  of  ab  yards  at  c  dollars  per  yard,  would  be 
represented  by  c  taken  ab  times,  that  is,  by  ab'Xc,  which  is  repre- 
sented by  abc. 

Art.  6'3'.  It  is  shown  in  "  Ray's  Arithmetic,"  Part  III,  Art.  44, 
that  the  product  of  two  factors  is  the  same,  whichever  be  made 
the  multiplier ;  we  will,  however,  demonstrate  the  principle  here. 

Suppose  we  have  a  sash  containing  a  vertical,  and  b  horizontal 
rows ;  there  will  be  a  panes  in  each  horizontal  row,  and  b  panes 
in  each  vertical  row  ;  it  is  required  to  find  the  number  of  panes  in 
the  window.  " 

It  is  evident,  that  the  whole  number  of  panes  in  the  window 
will  be  equal  to  the  number  in  one  row,  taken  as  many  times  as 
there  are  rows.  Then,  since  there  are  a  vertical  rows,  and  b 
panes  in  each  row,  the  whole  number  of  panes  will  be  represented 
by  b  taken  a  times,  that  is,  by  ab. 

Again,  since  there  are  b  horizontal  rows,  and  a  panes  in  each 
row,  the  whole  number  of  panes  will  be  represented  by  a  taken  b 
times,  that  is,  by  ba.     But,  since  either  of  the  expressions,  ba  or 

Review.— 65.  What  is  Multiplication  in  Algebra?  What  is  the  multi- 
plicand? The  multiplier?  The  product?  What  are  the  multiplicand  and 
multiplier  generally  called  ?  66.  How  do  you  multiply  a  literal  quantity 
by  a  number? 


48  RAY'S   ALGEBRA,    PART   FIRST. 


ab,  represents  the  whole  number  of  panes  in  the  window,  they  are 
equal  to  each  other,  that  is,  ab  is  equal  to  ba.  Hence,  it  follows, 
that  ihe  product  of  two  factors  is  the  same,  whichever  be  made  the 
midtiplier. 

By  taking  a=3  and  6=4,  the  figure  in  the  margin  may 
he  used  to  illustrate  the  principle  in  a  particular  case. 

In  the  same  manner,  the  product  of  three  or  more 
quantities  is  the  same,  in  whatever  order  they  are  taken. 
Thus,  2X3X4=3X2X4=4X2X3,  since  the  product 
in  each  case  is  24. 

1.  What  will  2  boxes,  each  containing  a  lemons,  cost  at  b  cents 
per  lemon  ? 

One  box  will  cost  ab  cents,  and  2  boxes  will  cost  twice  as  much 
as  1  box,  that  is,  2ab  cents. 

2.  What  is  the  product  of  26,  multiplied  by  3a? 

The  product  will  be  represented  by  26X3a,  or  by  3aX26,  or  by 
2X3X«6,  since  the  product  is  the  same,  in  whatever  order  the 
factors  are  placed.  But  2X3  is  equal  to  G,  hence  the  product 
26X3a  is  equal  to  6a6. 

Hence,  we  see,  that  in  multiplying  one  monomial  by  another, 
ihe  coefficient  of  ihe  product  is  obtained  by  midtiplying  together  the 
coefficients  of  the  multiplicand  and  multiplier.  This  is  termed,  the 
ride  of  the  coefficients. 

Art.  6§.  Since  the  product  of  two  or  more  factors  is  the  same, 
in  whatever  order  they  are  written,  if  we  take  the  product  of  any 
two  factors,  as  2X3,  and  multiply  it  by  any  number,  as  5,  the 
product  may  be  written  5X2X3,  or  5X3X2,  that  is,  10X3,  or 
15X2,  cither  of  which  is  equal  to  30.  From  which  we  sec,  that 
7vhen  either  of  the  factors  of  a  product  is  nniUipHed,  the  product 
itself  is  multiplied. 

Art.  69. — 1 .  What  is  the  product  of  a  by  a  ? 

The  product  of  b  by  a  is  written  ab,  hence,  the  product  of  a  by 
a  would  be  written  aa ;  but  this,  (Art.  33,)  for  the  sake  of  brevity, 
is  written  d^. 

2.  What  is  the  product  of  a^  by  a  ? 

Since  a^  may  be  written  thus,  aa,  the  product  of  a^  by  a  may  be 

Review. — 67.  Prove  that  3  times  4  is  the  same  as  4  times  3.  Prove 
that  «  times  h  is  the  same  as  h  times  o.  Is  the  product  of  any  number  of 
factors  changed  by  altering  their  arrangement?  In  multiplying  one  mono- 
mial by  another,  how  is  the  coefficient  of  the  product  obtained?  68.  If  you 
multiply  one  of  the  factors  of  a  product,  how  does  it  affect  the  product? 
69.  How  may  the  product  of  a  by  a  be  written  ?  How  may  the  product  of 
aa  by  a  be  written  ? 


MULTIPLICATION.  49 


expressed  thus,  aaX«>  or  aaa,  which,  for  the  sake  of  brevity,  is 
written  a^.  Hence,  the  exponent  of  a  letter  in  the  product,  is  equal 
to  the  sum  of  its  exponents  in  the  two  factors.  This  is  termed,  the 
rule  of  the  exponents. 

3.  What  is  the  product  of  a^  by  a"-^?  ....    Ans.  aaaa,  or  a*. 

4.  What  is  the  product  of  a'^b  by  a6  ?  .    .    Ans.  aaahh,  or  a^¥. 

5.  What  is  the  product  of  2ah^  by  3a6  ?  Ans.  Qaahbh,  or  Qd%^ 
Hence,  the 

RULE, 

FOR   MULTIPLYING   ONE    POSITIVE    MONOMIAL   BY    ANOTHER. 

Multiply  the  coefficients  of  the  two  terms  together,  and  to  their  pro- 
duct annex  all  the  letters  in  both  quantities,  giving  to  each  letter  an 
exponent  equal  to  the  sum  of  its  exponents  in  the  two  factors. 

Note. — It  is  customary  to  write  the  letters  in  the  order  of  the  alphabet. 
Thus,  a^Xc  is  generally  written  ahe. 

6.  Multiply  a6  by  a; Ans.  abx. 

7.  Multiply  26c  by  mn Ans.  2bcmn. 

8.  Multiply  4ab  by  5xy Ans.  20abxy. 

9.  Multiply  7ax  by  4:cd Ans.  28acdx. 

10.  Multiply  Gbij  by  Sax Ans.  18abxt/. 

11.  Multiply  3a'^6  by  4a& Ans.  12a36^ 

12.  Multiply  2xif  by  Sx'y Ans.  6ry. 

13.  Multiply  4a¥x  by  5ax^i/ Ans.  20a'I/x^i/. 

14.  What  is  the  product  ofSa^b'c  by  5a6V?  .    .  Ans.  15a*6*c*. 

15.  What  is  the  product  of  7xyh  by  Sx^i/z?    .    .  Ans.  56x*i/V. 

ISToTE. — The  learner  must  be  careful  to  distinguish  between  the  coeflS- 
eient  and  the  exponent.  Thus,  2a  is  different  from  a\  To  fix  this  in  his 
mind,  kt  him  answer  such  questions  as  the  following: 


What  is  2a— a'^  equal  to,  when  a  is  1  ? 


Ans.     1. 


What  is  a"'^ — 2a  equal  to,  when  a  is  5  ?      Ans.  15. 

What  is  a^ — 3a  equal  to,  when  a  is  4  ?      Ans.  52. 

What  is  a*— 4a  equal  to,  when  a  is  3  ?      Ans.  69. 

Art.  YO. — 1.  Suppose  you  purchase  5  oranges  at  4  cents  a 
piece,  and  pay  for  them,  and  then  purchase  2  lemons  at  the  same 
price ;  what  will  be  the  cost  of  the  whole  ? 

5  oranges,  at  4  cents  each,  will  cost  20  cents ;  2  lemons,  at  4 
cents  each,  will  cost  8  cents,  and  the  cost  of  the  whole  will  be 
20+8=28  cents. 

The  work  may  be  written  thus :     5+2 

4 

20+8=28  cents. 
5 


i>0  RAY'S    ALGEBRA,    FART   FIRST. 

If  you  purchase  a  oranges  at  c  cents  a  piece,  and  h  lemons  at  c 
cents  a  piece,  what  will  be  the  cost  of  the  whole  ? 

The  cost  of  a  oranges,  at  c  cents  each,  will  be  ac  cents ;  the 
cost  of  h  lemons,  at  c  cents  each,  will  be  he  cents,  and  the  whole 
cost  will  be  ac-\-hc  cents. 
The  work  may  be  written  thus :     a-\-h 

c 
ac-\-hc 
Hence,  when  the  sign  of  each  term  is  positive,  we  have  the 
following 

RULE, 

FOR    MULTIPLYING   A    POLYNOMIAL    BY    A    MONOMIAL. 

Multiply  each  term  of  the  multiplicand  hy  the  multiplier. 
EXAMPLES. 

2.  Multiply  a-\-d  by  6 Ans.  ab+hd. 

3.  Multiply  ac+6c  by  c?. An^.  acd-\-hcd. 

4.  Multiply  4x+5?/ by  3a Ans.  12ax+15ay. 

5.  Multiply  2x+2z  by  26 Ans.  4tbx+Qhz. 

6.  Multiply  m-\-2n  by  Zn Ans.  Smn-\-Qn'. 

7.  Multiply  ic+?/  by  ax Ans.  ax^-\-axy. 

8.  Multiply  x^-{-y'^  by  xy Ans.  i^y^xif. 

9.  Multiply  2x^by  by  ahx Ans.  2abx'^-^5abxy. 

10.  Multiply  3x'+2xz  by  2xz Ans.  6r%+4xV. 

11.  Multiply  3a+26+5c  by  4d  .    .    .  Ans.  12ari+86c;+20cdr. 

12.  Multiply  bc-\-af-{-mx  by  3ax.    .  Ans.  Sabcx^Sayx-]-3amx\ 

13.  Multiply  aft+ax+xy  by  aftxy.  .  AwQ.a^Jj^xy-^a^bxhj-^-abx^y^. 
Art.  71. — 1.  Let  it  be  required  to  find  the  product  of  x+y  by 

a-f  6.  Here  the  multiplicand  is  to  be  taken  as  many  times  as 
there  are  units  in  a+6,  and  the  whole  product  will  evidently  be 
equal  to  the  sum  of  the  two  partial  products.     Thus, 

x+y 

a+b 

ax+ay— the  multiplicand  taken  a  times. 

6a;+6y=:the  multiplicand  taken  b  times. 
ax-\-ay-\-hx-{-by=^i\\Q  multiplicand  taken  [a-\-b)  times. 

If  a:=.5,  y=Q,  a=2,  and  6=3,  the  multiplication  may  be  arranged 
thus :     5+6 
2+3 
10+12=the  multiplicand  taken  2  times. 

15+18=the  multiplicand  taken  3  times. 
10+27+18=55=the  multiplicand  taken  5  times. 


MULTIPLICATION.  51 

Hence,  when  all  the  terms  in  each  are  positive,  we  have  the 
following 

RULE, 

FOR    MULTIPLYING    ONE    POLYNOMIAL    BY    ANOTHER. 

Multiply  each  temn  of  the  multiplicand  by  each  term  of  the  multi- 
plier, and  add  the  products  together. 

2.  3. 

a+b  a'b^cd 

a+b  ab^cd^ 

d'-\-ab  d%''^abcd 

ab+¥  ■^d'bcd^'^c'^d? 

d'+2ab+b'  a^b''+a''bcd'+abcd'^'^^ 

4.  Multiply  a-i-b  by  c-\-d Ans.  ac-^ad-i-bc-\-bd. 

5.  Multiply  2x+SyhySa+2b.  .    .  Ans.  6ax+9ay+4bx^6by. 

6.  Multiply  2a+36  by  Sc-\-d.    .    .  Ans.  6ac-j-mc+2ad+Sbd. 

7.  Muliply  m-j-n  by  x-\-z Ans.  7nx-\-nx~\-mz-\-nz. 

8.  Multiply  4a+36  by  2a+b Ans.  Sd'+lOabi-Sb''. 

9.  Multiply  4x+  by  by  2a-^3x.  Ans.  Sax+ 1 0«?/+ 1 2x'^+ 1  day. 

10.  Multiply  3x+2?/  by  2x-^Sy Ans.  6x^+l3x7j+6y\ 

11.  Multiply  a^+fe^  by  a+6, Ans.  a^-\-d'b-^ab^+b\ 

12.  Multiply  3a'^+262  by  2a24-36^    .    .  Ans.  Ga*+l3aW+6bK 

13.  Multiply  a^+ab+b^  by  a-\-b.    .    .    Ans.  a^^2a''b-{-2ab^-^b\ 

14.  Multiply  c^+d^  by  c-\-d Ans.  c*^cd'+c^d+d*. 

15.  Multiply  x^-{-2xy+y^  by  x+y.  .    .  Ans.  3^-j-Sx^y-\-3xy^-\-y^. 

SIGNS. 

Art.  '72.  In  the  preceding  examples,  it  was  assumed  that  the 
product  of  two  positive  quantities,  is  also  positive.  It  ma}',  how- 
ever, be  shown  as  follows : 

1st.  Let  it  be  required  to  find  the  product  of  -f  ^  hy  a. 

The  quantity  b,  taken  once,  is  +5 ;  taken  twice,  is  evidently, 
4-26  ;  taken  3  times,  is  -\-3b,  and  so  on.  Therefore,  taken  a  times, 
it  is  -\-ah.  Hence,  the  product  of  two  positive  quantities  is  posi- 
tive ;  or,  as  it  may  be  more  briefly  expressed,  plus  multiplied  by 
plus,  gives  phis. 

2d.  Let  it  be  required  to  find  the  product  of  — b  ])y  a. 

Review. — To  what  is  the  exponent  of  a  letter  in  the  product  equal  ? 
What  is  the  rule  for  multiplying  one  positive  monomial  by  another !  70. 
What  is  the  product  of  a  plus  6,  by  c  ?  When  all  the  terms  in  each  are  posi- 
tive, how  do  you  multiply  a  polynomial  by  a  monomial  ?  71.  When  all 
the  terms  in  each  are  positive,  how  do  you  find  the  product  of  two  poly- 
nomials? 


52  RAY'S   ALGEBRA,    PART   FIRST. 

The  quantity  — h,  taken  once,  is  — h ;  taken  twice,  is  — 26 ; 
taken  3  times,  is  — 36 ;  and  hence,  taken  a  times,  is  — ah;  that  is., 
a  negative  quantity  multiplied  by  a  positive  quantity,  gives  a  nega- 
tive product.  This  is  generally  expressed,  by  saying,  that  minus 
multiplied  by  plus,  gives  minus. 

3d.  Let  it  be  required  to  multiply  h  by  — a. 

Since,  when  two  quantities  are  to  be  multiplied  together,  either 
may  be  made  the  multiplier  (Art.  67),  this  is  the  same  as  to  multi- 
ply — a  by  h,  which  gives  — ah.  That  is,  a  positive  quantity  mul- 
tiplied by  a  negative  quantity,  gives  a  negative  product ;  or,  more 
briefly,  plus  multiplied  by  minus,  gives  minus. 

4th.  Let  it  be  required  to  multiply  — 3  by  —2. 

The  negative  multiplier  signifies,  that  the  multiplicand  is  to  be 
taken  positively,  as  many  times  as  there  are  units  in  the  multi- 
plier, and  then  subtracted.  The  product  of  — 3  by  +2  is  — 6, 
then,  changing  the  sign  to  subtract,  the  — 6  becomes  -\-Q ;  and,  in 
the  same  manner,  the  product  of  — h  by  — a  is  -\-ab. 

Hence,  the  product  of  two  negative  quantities  is  positive ;  or, 
more  briefly,  minus  multiplied  by  minus,  gives  plus. 

Note. — The  following  proof  of  the  last  principle,  that  the  product  of 
two  negative  quantities  is  positive,  is  generally  regarded  by  mathematicians 
as  more  satisfactory  than  the  preceding,  though  it  is  not  quite  so  simple. 
The  instructor  can  use  either  method. 

5th.  To  find  the  product  of  two  negative  quantities 

To  do  this,  let  us  find  the  product  of  c — d  by  a — 6. 

Here  it  is  required  to  take  c — d  as  many  times  as  there  are  units  in  a — h. 
It  is  obvious  that  this  will  be  done  by  taking  c — d  as  many  times  as  there 
are  units  in  o,  and  then  subtracting  from  this  product,  c — d  taken  as  many 
times  as  there  are  units  in  h. 

Since  plus  multiplied  by  plus  gives  plus,  and  minus  multiplied  by  plus 
gives  minus,  the  product  of  c — d  by  a,  is  ac — ad. 

In  the  same  mannei',  the  product  of  c — d  by  h,  is  6c — hd',  changing  the 
signs  of  the  last  product  to  subtract  it,  it  becomes  — bc-\-hd;  hence  the  pro- 
duct of  c — d  by  a — h,  is  «c — ad — hc-\-hd. 

But  the  last  term,  -\-bd,  is  the  product  of — d  by  — h,  hence  the  product 
of  two  negative  quantities  is  positive  ;  or,  more  briefly,  minus  multiplied  by 
minun  produces  plit/i. 

The  multiplication  of  c — d  by  a — b  may  be  written  thus : 
c—d 
a—b 


ac — ad=c — d  taken  a  times. 

— bc-{-bd=c — d  taken  b  times,  and  then  subtracted. 
ac — ad — bc-\-bd 


MULTIPLICATION.  53 


Tho  operation  may  bo  illustrated  by  figures  j  thus,  let  it  bo  required  to 
find  the  product  of  7—4  by  5—3. 

7 — 4  We   first  take   5  times  7 — 4;   this  gives  a  product  too 

5 — 3  groat,  by  3   times  7 — 4,  or  21 — 12,  which,  being  subtracted 

35 — 20  fi'oni  the  first  product,  gives  for  the  true  result,  35 — 41-1-12^ 

— 214-12  which  reduces  to  4-6'     This  is  evidently  correct,  for  7 — 4 

35 41-}-12  =3,  and  5 — 3=2,  and  the  product  of  3  by  2  is  6. 

From  the  preceding  illustrations,  we  derive  the  following 

GENERAL   RULE, 

FOR    THE    SIGNS. 

Plus  multiplied  by  plus,  or  minus  muliiplied  hy  minus,  gives  plus. 
Plus  multiplied  hy  minus,  or  minus  midtiplied  hy  plus,  gives  minus. 
Or,  the  product  of  like  signs  gives  plus,  and  of  unlike  signs  gives  minus. 

From  all  the  preceding,  we  derive  the 

GENERAL  RULE, 

FOR    THE    MULTIPLICATION    OF    ALGEBRAIC    QUANTITIES. 

Midtiply  every  term  of  the  multiplicand,  hy  each  term  of  the  mul- 
tiplier.     Ohserving, 

1  St.  That  the  coefficient  of  any  term  is  equal  to  the  product  of  the 
coeffic  ients  of  its  factors. 

2d.  That  the  exponent  of  any  letter  in  the  product  is  equal  to  the 
sum  of  its  exponents  in  the  two  factors. 

3d.  That  the  product  of  like  signs,  gives  plus  in  the  product,  and 
unlike  signs,  gives  rninus.  Then,  add  the  several  partial  prodticts 
together. 

NUMERICAL  EXAMPLES, 

TO    VERIFY    THE    RULE    OF    THE    SIGNS. 

1.  Multiply  8-3  by  5 Ans.  40-15-^25=5X5. 

2.  Multiply  20-13  by  4 Ans.  80-52=28=7X4. 

3.  Multiply  13-7  by  11-8  .  Ans.  143-181+56=18=6X3. 

4.  Multiply  10^  3  by  3-5. 

Ans.  30-41~15=-26=13X-2. 

5.  Multiply  9-5  by  8-2.   .    .    Ans.  72-58+10=24=4X6. 

6.  Multiply  8-7  by  5-3.   .    .    .  Ans.  40-59+21=2=1x2. 

Review.— 72.  What  is  the  product  of +?>  by +o?  Why?  What  is  tho 
product  of  — 6  by  a  ?  Why  ?  What  is  the  product  of  -\-h  by  —a  ?  Why  ? 
What  is  the  product  of  —3  by  — 2  ?  What  does  a  negative  multiplier  sig- 
nify? What  does  minus  multiplied  by  minus  produce?  What  is  the  gen- 
eral rule  for  the  signs  ?  What  is  the  general  rule  for  the  multiplication  of 
algebraic  quantities  ? 


54  RAY'S    ALGEBRA,    PART    FIRST. 


GEi\ERAL  EXAMPLES. 

1.  Multiply  Sa\x7/  by  7axi/^ Ans.  21aV?/*. 

2.  Multiply —Sa^i  by  3a6"' Ans. —I5a'b*. 

3.  Multiply  —5x\ij  by  —5a;/ Ans.  25a^y\ 

4.  Multiply  3«— 26  by  4c Ans.  I2ac—Sbc. 

5.  Multiply  3a:+2y  by  —2x Ans.  --6x'—4xi/. 

6.  Multiply  a-\-b  by  x — y Ans.  ax — ay-^bx — by. 

7.  Multiply  a—b  by  a— 6 Ans.  d^—2ab-\-b'K 

8.  Multiply  d^Arac+c'-  by  a— c Ans.  o?—c^. 


9.  Multiply  m+?i  by  m — ii An 


s.  nr — n 


10.  Multiply  a2—2a6+/;-' by  a+&.    .    .    .  A.n^.a^~d'b—a¥-^b^. 

11.  Multiply  *Sxhj—2xy^-^y''  by  2xy\yK 

Ans.  Gx-y + 3.xV— 4^y +/• 

12.  Multiply  a2+2a&+&'  l3y  «'— 2a6+6-.   .   Ans.  \&~2d'b'^b\ 

13.  Multiply  ?/■''— ?/+l  by  y+l Ans.  y'+l. 

14.  jNIultiply  x'^+J/^  by  x^ — \f Ans.  x^ — ?/*. 

15.  Multiply  a--3a+8  by  a4-3 Ans.  a=^— a+24. 

16.  Multiply  2x^-'^xy^f  by  a;^— Sx//. 

Ans  2a;^—  1 3x5?/  + 1  Qxhf—hxiJ\ 

17.  Multiply  3a+56  by  3a— 56 Ans.  9«'''-25);-. 

18.  Multiply  2a'^— 4ax+2x-  by  3a -3a:. 

Ans.  (Ja''- 1 8a-a:+ 1 8ax2— Grl 

19.  Multiply  5x''+3y' by  5x^-3^'' Ans.  25x«-    9//. 

20.  Multiply  2a'+2a-x+2ax2+2x''  by  3a-3.T.  .  Ans.  Ga*— fu^ 

21.  Multiply  3a2+3ax+3x2  by  2a2-2ax.    .    .     Ans.  Ga*- Gaxl 

22.  Multiply  3«''^+5ax— 2x'^  by  2a— x. 

Ans.  Ga=^+7a2x-9ax2+2xl 

23.  Multiply  x«+x*+x2  by  x2—l Ans.  x«— xl 

24.  Multiply  x^Arxy-\-rf  by  x^ — xy^y'-.      .    .    Ans.  x*-^x'^y'^-\-y*. 

25.  Multiply  a^-i-d'b^ab''+P  by  a—b Ans.  a*—h*. 

In  the  following  examples,  let  the  pupil  perform  the  multipli- 
cations indicated,  by  multiplying  together  the  quantities  contained 
in  the  parentheses. 

26.  (x— 3)(x-3)(x-3).  ......     Alls.  x3-9x2+27x-27. 

27.  (x— 4)(x— 5)(x+4)(x+5) Ans.  x*-41x'^+400. 

28.  (a4-c)(a— c)(a4-c)(a— c) Ans.  a*— 2aV+c*. 

29.  {d'+b'-^c'—ab—ac—bc){ai-b+c).  .    Ans.  a^+b^-j-c'—Sabc. 
dO.  {n'+n+l){n'+n'\-l){n—l){n—l).     .    .    Ans.  n''-2n^+i. 


DIVISION.  55 


DIVISION. 

Art.  vs.  Division  in  Algebra,  is  the  process  of  finding  how 
often  one  algebraic  quantity  is  contained  in  another. 

Or,  it  may  be  defined  thus :  Having  the  product  of  two  factors,  Jind 
one  of  them  given,  Division  teaches  the  method  of  finding  the  other. 

The  number  by  which  we  divide,  is  called  the  divisor ;  the  num- 
ber to  be  divided,  is  called  the  dividend;  the  number  of  times  the 
divisor  is  contained  in  the  dividend,  is  called  the  quo/ient. 

Art.  74.  Since  Division  is  the  reverse  of  Multiplication,  the 
quotient,  multiplied  by  the  divisor,  must  produce  the  dividend. 

The  usual  method  of  indicating  Division,  is  to  write  the  divisor 
under  the  dividend  in  the  form  of  a  fraction.     Thus,  to  indicate 

that  ab  is  to  be  divided  by  a,  we  write,  — .     Algebraic  Division, 

however,  is  sometimes  indicated,  like  that  of  whole  numbers,  thus, 
a)ab ;  where  a  is  the  divisor,  and  ab  the  dividend. 

Note  to  Teachers. — In  solving  the  following  examples,  let  the 
pupil  give  the  reason  for  the  answer,  as  in  the  solution  to  the  first  question. 
Although  the  examples  can  be  solved  mentally,  it  will  be  found  most  advan- 
tageous, to  work  them  on  the  slate,  or  blackboard;  as  the  learner,  by  this 
means,  will  be  preparing  for  the  performance  of  more  difficult  operations. 

4x 

1.  How  often  is  X  contained  in  4x? Ans.  — ~4. 

X 

This  solution  is  to  be  given  by  the  pupil,  thus:  4a: divided  byx, 
is  equal  to  4,  because  the  product  of  4  by  x  is  4x. 

2.  How  often  is  a  contained  in  6a  ? Ans.  6. 

3.  How  often  is  a  contained  in  a6? Ans.  &. 

4.  How  often  is  b  contained  in  ^ab'l Ans.  3a. 

5.  How  often  is  a  contained  in  ahxt      ......      Ans.  bx. 

6.  How  often  is  a  contained  in  fiabxt Ans.  56x. 

7.  How  often  is  2  contained  in  4a? Ans.  2a. 

8.  How  often  is  2a  contained  in  Aah  ? Ans.  26. 

9.  How  often  is  a  contained  in  a/^? Ans.  a. 

10.  How  often  is  a  contained  in  a^? Ans.  a^ 

11.  How  often  is  a  contained  in  3a"^? Ans.  3a. 

12.  How  often  is  ab  contained  in  5a-6? Ans.  5a. 

R  E  V I E  w. — 73.  What  is  Algebraic  Division  ?  What  is  the  divisor  ?  Tho 
dividend?  The  quotient?  74.  To  what  is  the  product  of  the  quotient  .and 
the  divisor  equal  ?  Why?  What  is  the  usual  method  of  indicating  divi- 
sion ? 


56  RAY'S   ALGEBRA,    PART   FIRST. 

13.  How  often  is  2a    contained  in  lOa^? Ans.  5a^. 

14.  How  often  is  3a^    contained  in  12a^5?    ....    Ans.  4a6. 

15.  How  often  is  4a6^  contained  in  12a^6^c?     .    .      Ans.  3a^6c. 

16.  How  often  is  2a^    contained  in  6a^b? 

Solution,  -r——^=~a^-^b=3a^b. 
2a^     2 

In  obtaining  this  quotient,  we  readily  see, 

1st.  The  coefficient  of  the  quotient,  must  be  such  a  number, 
that  when  multiplied  by  2,  it  shall  produce  6 ;  hence,  to  obtain  it, 
we  divide  6  by  2. 

2d.  The  exponent  of  a  must  be  such  a  number,  that  when  2,  the 
exponent  of  a  in  the  divisor,  is  added  to  it,  the  sum  shall  be  5 ; 
hence,  to  obtain  it,  we  must  subtract  2  from  5;  that  is,  5—2  is 
equal  to  3,  the  exponent  of  a  in  the  quotient. 

3d.  The  letter  b,  which  is  a  factor  of  the  dividend,  but  not  of 
the  divisor,  must  be  found  in  the  quotient,  in  order  that  the  product 
of  the  divisor  and  quotient  may  equal  the  dividend. 

Art.  75.  It  remains  to  ascertain  the  rule  for  the  signs. 

Since  +a  multiplied  by  +6= 
plus  divided  by  plus,  gives  plus 

Since  — a  multiplied  by  +6= 
minus  divided  hy  plus,  gives  minus. 

Since  -{-a  multiplied  by  — b= — a 
minus  divided  by  minus,  gives  plus. 

Since  — a  multiplied  by  —b=-\-ab,  therefore,  — ~= — a;  hence, 
plus  divided  by  minus,  gives  minus. 

From  this,  we  see,  that  in  Division,  like  signs  give  plus,  and 
unlike  signs  give  minus. 

Hence  the 

RULE, 

FOR    DIVIDING    ONE    MONOMIAL   BY    ANOTHER. 

Divide  the  coefficient  ofilie  dividend,  by  that  of  the  divisor  ;  observ- 
ing, that  like  signs  give  plus,  and  unlike  signs  give  minus. 

After  the  coefficient,  ivrite  the  letters  common  to  both  divisor  and 
dividend,  giinng  to  each  an  exponent,  equal  to  the  excess  of  the  expo- 
nent of  the  same  letter  in  the  dividend,  over  that  in  the  divisor. 

In  the  quotient,  ivrite  the  letters  with  their  respective  exponents,  that 
are  found  in  the  dividend,  but  not  in  the  divisor. 


Since  +a  multiplied  by  -|-6=+a?>,  therefore.  — --=-fa;  hence, 
Since  —a  multiplied  by  -\-b=—ab,  therefore,  -T7-=— «;  hence, 
Since  -{-a  multiplied  by  — b= — ab,  therefore,  — j-=-\-a;  hence, 


DIVISION.  57 


Note. — Tho  i^upil  must  recollect,  that  when  a  letter  has  no  exponent 
expressed,  1  is  understood ;  thus,  a,  is  the  same  as  a'. 

EXAMPLES. 

17.  Divide  15a^6c  by  3a26 Ans.  5ac. 

18.  Divide  '^Ixhf  by  — 3xy Ans.  — Ox?/. 

19.  Divide —18a^.r  by —Gax Ans.  3a^ 

20.  Divide  ^bahxij  by  bmj Ans.  bbx. 

21.  Divide  aV  by  d^x Ans.  aV. 

22.  Divide  —4,a^x'  by  2a^x Ans.  -2d'x. 

23.  Divide  — 126*xy  by  —Ac'xif Ans.  'Sx'if. 

24.  Divide  — 24aV?/^y  by  Ad^xyv Ans.  — 6aV-?/. 

25.  Divide  Qacx-ifv  by  ^ax^y^v Ans.  2cyK 

26.  Divide  — \Oc^x''y^v  by  — 2cy^v Ans.  ^cxhj. 

27.  Divide  ma-'xhj-  by  12^^ Ans.  5aVy. 

28.  Divide —18aV-W  by —6aViJ^ Ans.  3c W. 

29.  Divide  ~2Sac^x^y'v'  by  14axy Ans.  —2c'x^vK 

30.  Divide  30a6-Vxy  by  — 2aex* Ans.  —Ibc'ehf. 

Note.— Although  the  method  of  operation  in  each  of  the  following  ex- 
amples is  the  same  as  in  the  preceding,  they  may  be  passed  over,  until  the 
book  is  reviewed. 

31.  Divide  {x-{-yY  by  {x-\-y) Ans.  {x-\-y). 

32.  Divide  {a-\-hf  by  («+&)' Ans.  (a+6). 

33.  Divide  («+&)*  by  (a+&) Ans.  (a+Sf. 

34.  Divide  Q{m^nY  by  2(m+w) Ans.  3(w+«)2. 

35.  Divide  d\h-\-cY  by  a(6+c) Ans.  «(6+c). 

36.  Divide  Qd'h[x+yY  by  2a6(x4-?/)' Ans.  3«(x+?/). 

37.  Divide  [x+y){a—hf  by  [a—h).      .    .    .  Ans.  {x+y)[a—h)\ 

38.  Divide  [x — yY[m — ny  by  (x — yY{m — nY-  .    .    Ans.  [x—t/]. 

Art.  76.  It  is  evident,  that  one  monomial  cannot  be  divided 
by  another,  in  the  following  cases. 

1st.  AVhen  the  coefficient  of  the  dividend  is  not  exactly  divisible 
by  the  coefficient  of  the  divisor. 

2d.  When  the  same  literal  factor  has  a  greater  exponent  in  the 
divisor  than  in  the  dividend. 

3d.  "When  the  divisor  contains  one  or  more  literal  factors,  not 
found  in  the  dividend. 

In  each  of  these  cases,  the  division  is  to  be  indicated  by  Avriting 
the  divisor  under  the  dividend,  in  the   form  of  a  fraction.     The 

Review. — 75.  When  the  signs  of  the  dividend  and  divisor  are  alike, 
what  will  be  the  sign  of  the  quotient?  Why  ?  When  the  signs  of  the  divi- 
dend and  divigor  are  unlike,  what  will  be  the  sign  of  the  quotient?  Why  ? 
What  is  the  rule  for  dividing  one  monomial  by  another? 


58  RAY'S    ALGEBRA,    PART   FIRST. 

fraction  thus  found,  may  often  be  reduced  to  lower  terms.     For 
the  method  of  doing  this,  see  Art.  129. 

Art.  "77.  It  has  been  shown,  in  Art.  68,  that  any  product  is 
multiplied,  by  multiplying  either  of  its  factors  ;  hence,  conversely, 
any  dividend  will  be  divided,  by  dividing  either  of  its  factors. 

Thus,  -^=2X6=12,  by  dividing  the  factor  4. 

Or,  1^=4X3^12,  by  dividing  the  factor  6. 

DIVISIOIV  OF  A  POLYNOMIAL  BY  A  MOXOML^L. 

Art.  "78.  Since,  in  multiplying  a  polynomial  by  a  monomial, 
we  multiply  each  term  of  the  multiplicand  by  the  multiplier; 
therefore,  we  have  the  following 

RULE, 

FOR    DIVIDING    A    POLYNOMIAL    BY    A    MONOMIAL. 

Divide  each  term  of  the  dividend,  by  the  divisor,  according  to  the 
rule  for  the  division  of  monomials. 

EXAMPLES. 

1.  Divide6a:+12//by  3 Ans.  2a:+4y. 

2.  Divide  15.r;— 206  by  5 Ans.  3x— 46. 

3.  Divide21a+356by— 7 Ans.  — 3a— 56. 

4.  Divide  Gax+9ay  by  3a Ans.  2x+Sy. 

5.  Divide  a6+ac  by  a Ans.  6+c.- 

6.  Divide  a6c — acfhy  ac Ans.  b—f. 

7.  Divide  12a?/— Sac  by —4a Ans.  — 3y+2c. 

8.  Divide  lOax — 15a?/  by  —5a Ans.  —2x-\-Sy. 

9.  Divide  126a.— 18x'-^  by  6x Ans.  26— 3x. 

10.  Divide  a'^6'^-2a6='x  by  a6 Ans.  a6— 26^0:. 

11.  Divide  12a-6c — 9a6'x^+6a6'^6'  by  — 3ac. 

Ans.  —4a6+ 3x2-261 

12.  Divide  1 5a^b''c— 21  a'^bh^  by  Sa'^bc.   .    .    .      Ans.  5a''6— 76^^. 

13.  Divide  6a^6c+2a"''6c'^— 4aV  by  2a'-^c.    .     Ans.  3a6-!-6c— 26'^. 

Note. — The  following  examples  may  be  omitted  until  the  book  is  reviewed. 

14.  Divide  6(a+c)+9(a+a;)  by  3.     .    .  Ans.  2(a+c)+3(a+x). 

15.  Divide  5a(x+?/) — 10a''^(a; — ij)  by  5a.  Ans.  {x-{-y)—2a{x — y). 

16.  Divide  d'b{c+d)+ab''{c^—d)  by  a6.    Ans.  a{c+d)+b{c'—d). 

Re  VI  E  w^ — 76.  In  what  case  is  the  exact  division  of  one  monomial  by 
another  impossible  ?  78.  What  is  the  rule  for  dividing  a  polynomial  by  a 
monomial  ? 


DIVISION.  59 


17.  Divide  ac(w+w)—6c(?/i+n)  by  w-f?i Ans.  ac~bc. 

18.  Divide  l2{a—bY+6c{a—bfhy2{a—b). 

Ans.  G{a—b)+Sc{a—b)\ 

19.  Divide  2d'c{x+j/Y+2ac\x+i/y  by  2ac{x^i/Y. 

Ans.  «(a;+y)+c(x+y)^ 

20.  Divide  {m-i-u){xi-7/Y+{m4-n){x—7/Y  by  wi+n. 

Ans.  (a:-fy)2+(x— 1/)2. 

DIVISION    OF    Oi\E    POLYNOMIAL    BY    AIVOTHER. 

Art.  '79.  To  explain  the  method  of  dividing  one  polynomial  by 
another,  Ave  may  regard  the  dividend  as  a  product,  of  which  the 
divisor  and  the  quotient  are  the  two  factors.  We  shall  examine 
the  method  of  forming  this  product,  and  then,  by  a  reverse  opera- 
tion, explain  the  process  of  division. 


Multiplication,  or  formation  of  a  product. 

2d~ — ab 
a — b 


2a^--a% 

—2a''b-\-ab' 

Ua^—Sd'b+ab' 


Division,  or  decomposition  of  a  product. 

2a^—Sa''b+ab''\a—b 

2a?-2a'b         2o}-ab 

1st.  rem.        —d^b+ab'^ 
~a'b+ab^ 


2d.  rem.  0 


If  we  multiply  2a^ — ab  by  a — b,  and  arrange  the  terms  according 
to  the  powers  of  a,  we  shall  find  the  product  to  be  2a^ — Sd^b-\-ab^. 

In  this  multiplication  w^e  remark, 

1st.  Since  each  term  in  the  multiplicand  is  multiplied  by  each 
term  in  the  multiplier,  if  no  reduction  takes  place  in  adding  the 
several  partial  products  together,  the  number  of  terms  in  the  final 
product  will  be  equal  to  the  number  produced  by  multiplying  to- 
gether the  number  of  terms  in  the  two  factors.  Thus,  if  one  fac- 
tor have  3  terms,  and  the  other  2,  the  number  of  terms  in  the 
product  will  be  six.  Frequently,  however,  a  reduction  takes  place, 
by  which  the  number  of  terms  is  lessened.  Thus,  in  the  above 
example,  two  terms  being  added  together,  there  are  only  3  terma 
in  the  product. 

2d.  In  every  case  of  multiplication,  there  are  two  terms  which 
can  never  be  united  with  any  other.  These  are,  first:  that  term 
which  is  the  product  of  the  two  terms  in  the  factors,  which  contain 
the  highest  power  of  the  same  letter;  and  second:  the  term  which 
is  the  product  of  the  two  terms  in  the  factors,  which  contain  the 
Jowest  power  of  the  same  letter. 

From  the  last  principle  it  follows,  that  if  the  term  containing 
the  highest  power  of  any  letter  in  the  dividend,  be  divided  by  the 
term  containing  the  highest  power  of  the  same  letter  in  the  divisor, 


60  RAY'S   ALGEBRA,    PART    FIRST. 

the  result  will  be  the  term  of  the  quotient  containing  the  high- 
est power  of  that  letter.  Hence,  if  2a^  be  divided  by  a,  the  result, 
2d\  will  be  the  term  containing  the  highest  power  of  a,  in  the 
quotient. 

The  dividend  expresses  the  sum  of  the  partial  products  of  the 
divisor,  by  the  different  terms  of  the  quotient.  If,  then,  we  form 
the  product  of  the  divisor  by  the  first  term,  2a^  of  the  quotient, 
and  subtract  it  from  the  dividend,  the  remainder,  — a^b-]-ah^,  Avill 
be  the  sum  of  the  other  partial  products  of  the  divisor,  by  the 
remaining  terms  of  the  quotient. 

Now,  since  this  remainder  is  produced,  by  multiplying  the  divi- 
sor by  the  remaining  terms  of  the  quotient,  it  follows,  as  in  the 
method  of  obtaining  the  first  term  of  the  quotient,  that  if  the  term 
containing  the  highest  power  of  a  particular  letter  in  this  remain- 
der, be  divided  by  the  term  containing  the  highest  power  of  the 
same  letter  in  the  divisor,  the  quotient  will  be  the  term  containing 
the  highest  power  of  that  letter  in  the  remaining  terms  of  the 
quotient. 

Hence,  if  — a^b  be  divided  by  a,  the  quotient,  — ab,  will  be  an- 
other term  of  the  quotient.  Multiplying  the  divisor  by  this  second 
term,  and  subtracting,  we  find  the  second  remainder  is  0 ;  hence, 
the  exact  quotient  is  2a^ — ab.  Had  there  been  a  second  remain- 
der, the  third  term  of  the  quotient  would  have  been  obtained  from 
it  in  the  same  manner  as  the  second  term  was  obtained  from  the 
first  remainder. 

Since  each  term  of  the  quotient  is  found,  by  dividing  that  term 
of  the  dividend  containing  the  highest  power  of  a  particular  letter, 
by  the  term  of  the  divisor  containing  the  highest  power  of  the  same 
letter,  it  is  more  convenient  to  place  the  terms  of  the  dividend  and 
the  divisor,  so  that  the  exponents  of  the  same  letter  shall  either 
increase  regularly,  or  diminish  regularly,  from  the  left  to  the  right. 
This  is  termed,  arranging  the  dividend  and  divisor,  with  reference  to 
a  certain  letter.  The  letter  with  reference  to  which  a  quantity  is 
arranged,  is  called  the  letter  of  arrangement. 

The  divisor  is  placed  on  the  right  of  the  dividend,  because  it  is 
more  easily  multiplied  by  the  respective  terms  of  the  quotient,  as 
they  are  found. 

From  the  preceding,  we  derive  the 

RULE, 

FOR    THE    DIVISION    OF    ONE    POLYNOMIAL    BY    ANOTHER. 

Arrange  the  dividend  and  divisor,  with  reference  to  a  certain  letter, 
and  place  the  divisor  on  the  right  of  the  dividend. 


DIVISION.  61 


Divide  the  first  term  of  the  dividend  hy  the  first  term  of  the  divisor, 
the  residt  will  be  the  first  term  of  the  quotient.  Multiply  the  divisor 
by  this  term,  and  subtract  the  product  from  the  dividend. 

Divide  the  first  term  of  the  remainder  by  the  first  term  of  the  divi- 
sor, the  result  will  be  the  second  term  of  the  quotient.  Multiply  the 
divisor  by  this  term,  and  subtract  the  product  from  the  last  re- 
mainder. 

Proceed  in  the  same  manner,  and  if  you  obtain  Ofor  a  remainder, 
the  division  is  said  to  be  exact. 

Remark  s. — 1st.  It  is  not  absolutely  necessary  to  arrange  the  dividend 
and  divisor  with  reference  to  a  certain  letter;  it  should  always  be  done, 
however,  as  a  matter  of  convenience. 

2d.  The  divisor  may  be  plac^  on  the  left  of  the  dividend,  instead  of  the 
right,  as  directed  in  the  rule.  When  the  divisor  is  a  monomial,  it  is  more 
convenient  to  place  it  on  the  left;  but,  when  it  is  a  polynomial,  to  place  it 
on  the  right. 

3d.  If  there  are  more  than  two  terms  in  the  quotient,  it  is  not  necessary 
to  bring  down  any  more  terms  of  the  remainder,  at  each  successive  subtract- 
tion,  than  have  corresponding  terms  in  the  quantity  to  be  subtracted. 

4th.  It  is  a  useful  exercise  for  the  learner,  to  perform  the  same  example 
in  two  different  ways.  First,  by  arranging  the  dividend  and  divisor,  so 
that  the  powers  of  the  same  letter  shall  diminish  from  left  to  right ;  and, 
secondly,  so  that  the  powers  of  the  same  letter  shall  increase  from  left  to 
right. 

5th.  It  is  evident,  that  the  exact  division  of  one  polynomial  by  another 
will  be  impossible,  when  the  first  term  of  the  arranged  dividend  is  not 
exactly  divisible  by  the  first  term  of  the  arranged  divisor;  or,  when  the  first 
term  of  any  of  the  remainders  is  not  divisible  by  the  first  term  of  the 
divisor. 


Divide  6a'^— 13ar+6a:2  by  2a— Sx. 
6a:'—lSax+6x''\2a—Sx 


6a''—9ax 
—4ax-\-6x' 
—4ax+6x^ 

.  Divide  x'^ — y"^  by  x — y. 
x^—y^\x—y 

3a— 2x  Quotient. 

3.  Divide  a^+ar*  by  a+x. 
a^+a^\a+x 

x'—xy      x+y  Quotient. 

a^-\-a''x        a^—ax+x^'  Quot. 

xy-y' 
xy—tf 

-a'x+a^ 
—a'^x—ax^ 

+ax^-\-x^ 
ax'+a^ 

62  RAY'S    ALGEBRA,    PART    FIRST. 

4.  Divide  5a'^x-{-5ax^-{-a'-i-x^  by  Aax-^-a^-^-xK 

a^+4a"^a:+ax'^  a'\-x  Quotient. 

a^xA;-4:ax-^3? 
a^x-\-4ax^-{-x^ 


In  this  example,  neither  divisor  nor  dividend  being  arranged 
with  reference  to  either  a  or  x,  we  arrange  tliem  with  reference  to 
a,  and  then  proceed  to  perform  the  division. 

5.  Divide  a''+a'—5a'+Sa^  by  «— a'^ 


Division  performed,  by  arranging 
both  quantities  according  to  the  as- 
cending powers  of  a. 

a^+a'—5a*+3a^\a—a' 


a-'-a' 
+2a'- 
•    2a'- 

a 

-2a' 

+2a'-Sa' 
Quotient. 

- 

-Sa'+Sa' 

-Sa'+Sa^ 

Division  performed,  by  arranging 
both  quantities  according  to  the  de- 
scending powers  of  a. 

Sa^—5a*+a'-\-a''\—a'^+a 

Sa^—Sa*  —Sa'+2a''+a 

— 2a* -T  a'  Quotient. 

-2a*-{-2a' 


-a^+a' 


The  pupil  will  perceive  that  the  two  quotients  are  the  same,  but  differently 
arranged. 

EXAMPLES. 

6.  Divide  4a^—8ax-i'4x^  by  2a— 2a: Ans.  2a— 2x. 

7.  Divide  2x2-f  7x?/+6/  by  x+2y Ans.  2x+3y. 

8.  Divide  2mx-{-Snx-\-l0mn-\-l5n'^  by  a;+5/t.    .  Ans.  2m-j-3n. 

9.  Divide  x^-{-2xy-\-y^  by  x-^ry Ans.  x-\-y. 

10.  Divide  8a*— 8^;'^  by  2a2— 2x2 Ans.  4a2-f  4x'^. 

11.  Divide  ac-j- 6c — ad — hdhj  a-\-h Ans.  c — d. 

12.  Divide  x^-\-r/-\-f)xy^^hx'^y  \ij  x^-^-Axy-^-y"^.    .    .    Ans.  x-ty. 

13.  Divide  a^— 9a^+27a— 27  by  a— 3.    .    .    .  Ans.  a'^— 6a+9. 

14.  Divide  4a* — 5aV+x*  by  2a''^ — 3ax+x''.   Ans.  2a'^^Sax^xK 

15.  Divide  x* — y*  by  x — y Ans.  x^-\-x^y-\-xy'^-]-y'. 

Review. — 79.  In  multiplying  one  polynomial  by  another,  what  terms 
in  the  product  cannot  be  added  together?  How  is  the  term  of  the  quotient 
found,  which  contains  the  highest  power  of  any  particular  letter?  After 
obtaining  the  first  remainder,  how  is  the  second  term  of  the  quotient  found  ? 
What  is  understood  by  arranging  the  dividend  and  divisor  with  reference 
to  a  certain  letter?  What  is  the  letter  of  arrangement?  Why  is  tho 
divisor  placed  on  the  right  of  the  quotient?  What  is  the  rule  for  tho 
division  of  one  polynomial  by  another  ?  When  is  the  exact  division  of  one 
polynomial  by  another  impossible  ? 


ALGEBRAIC    THEOREMS.  (53 

16.  Divide  o?—l/  by  a?-\-ah+¥ Ans.  a—h. 

17.  Divide  x^~i/+^xif-^xhj  by  x—y.       .    .    Ans.  x'-2x!/^-f, 

18.  Divide  4x*— 64  by  2x— 4.       .    .    .     Ans.  2x-3+4x-2+8x4-lG. 

19.  Divide  a^— 5a*x+10aV— lOa^x^+Sax"— x^  by  a'—2ax+x\ 

Ans.  a^ — Sa'-x-^Sax'^ — x^. 

20.  Divide  4a^—25a''x*+20a3f—4x^  by  2a^—5ax^-{-2r\ 

Ans.  2a='+5ax2-2x''. 

21.  Divide  2/3^1  by  y+1. Ans.y^—ij^l. 

22.  Divide  6a*+4a'x— 9aV— 3ax3+2x*  by  2a''+2ax—x\ 

Ans.  3a^ — ax— 2x-. 

23.  Divide  Sa'—8a''b'-^3a'c'+5b*-3b''c-'  by  a''-¥. 

Ans.  3a2_5&2-f  3c-^. 

24.  Divide  x«— 3xV+3xy— /  by  x-''— 3xV+3xy2— 7^. 

Ans.  x'^+3x^y+3xy^+y^- 

MISCELLANEOUS   EXERCISES. 

1.  3a+5x— 9c+7(^+5a— 3x— 3cZ-(4a+2x— 8c+4<^)=Avhat  ? 

Ans.  4a — c. 

2.  6ab~Scx+5d—ab+5cx—8d—{Sab+cx—Sd)=what ? 

Ans.  2ab-\-cx. 

3.  a+6— (2a— 36)— (5a+76)— (-13a+26)=what? 

Ans.  7a— 5b. 

4.  (a4-6)(a+&)  +  («— &)(«— &)=what?     ....  Ans.  2a2+26^ 

5.  {x-\-z){x-^z) — (x — z){x — 2)=what? Ans.  4xz. 

6.  (a2+a*+a«)(a2— 1)— (a*4-«)(a*— «)=what?  .    .    .      Ans.  0. 

7.  (a*+aV+z^)-^(a2— a2+22)— (a-f  2)(a— 2)=what?  j^  az-{-2z\ 
S.  {—l-^a^ii^)-^{—l-\-an)-\-{l-\-an){l—an)=whiit?  A.  2+a». 
9.  {a^+a-'b—ab^—b^)~{a—b)—{a—b){a—b)=what1  Ans.  4ab. 


CHAPTER  II. 

ALGEBRAIC    THEOREMS. 

DERIVED    FROM    MULTIPLICATION    AND    DIVISION. 

Art.  80.— If  we  square  a+b,  that  is,  multiply  a+fe  by  itself, 
the  product  will  be  a2-f-2a6+6'' ;  thus :  a-\-b 

a+b 


a^-[-a6 

■i-ab+b^ 
a''+2ab+b^ 


64  RAY'S    ALGEBRA,    PART   FIRST. 


But  a-\-b  is  the  sum  of  the  quantities,  a  and  b ;  hence 
THEOREM   I. 

The  square  of  the  sum  of  iico  quantities,  is  equal  to  the  square  of 
the  frst,  plus  twice  the  product  of  the  first  by  the  second,  plus  the 
square  of  the  second. 

EXAMPLES. 
Note. — The   instructor  should  read  each  of   the  following  examples 
aloud,  and  require  the  pupil,  by  applying  the  theorem,  to  write  at  once  the 
result  on  a  slate,  or  blackboard.     The  examples  may  be  enunciated  thus  : 
What  is  the  square  of  2a-\-h  ? 

1.  (2+3)'^=4+12+9=25 
j^  2.  (2a+6)"'=4a2+4a&+&'. 

4.  {ah-^cdf=a''h''-^2ahcd-^cH''. 

5.  {x^+xrjY^x^+2xhj+xh/. 

6.  (2a2+3ax)2=4a*4-12a3a;+9aV. 

Art.  81. — If  we  square  a — b,  that  is,  multiply  a — b  by  itself, 
the  product  will  be  a^ — 2a6-|-6^     Thus :  a — b 

a — b 
a^ — ab 

—ab+W 
a:'—2ab+W 
But  a — b  is  the  difference  of  the  quantities  a  and  6;  hence 

THEOREM   II. 

The  square  of  the  difference  of  two  quantities,  is  equal  to  the 
square  of  the  first,  minus  twice  the  product  of  the  first  by  the  second, 
plus  the  square  of  the  second.  ^ 

EXAMPLES. 

1.  (5-4)2=25—40+16=1. 

2.  [2a—bY=4:a^—^ab-^b\ 

3.  [^x~27jY=Qx'—\2xy+4y\ 

4.  [x^—y''Y=x'—2xhf-\-y\ 

5.  (ax— x2)2=aV— 2cfa;3+a;*. 

6.  (5a2-62)2=25a*— 10a2524.54, 

Art.  82. — If  we  multiply  a-\-b  by  a — b,  the  product  will  bo 
o'—bK     Thus:  a-\-b 
a — b 
a'^+ab 

—ab—b\ 
a^—b-" 


ALGEBRAIC    THEOREMS.  65 

But  a-{-b  represents  the  sum  of  two  quantities,  and  a — h,  their 

difference ;  hence, 

THEOREM    III. 

The  product  of  the  sum  and  difference  of  two  quantities,  is  equal 
to  the  difference  of  their  squai-cs. 

EXAiMPLES. 

1.  (5+3)(5-3)=25-9=lG-8X2. 

2.  {2a-{-b){2a—b)=4a'-h\ 

3.  {2x+Si/){2x—Si/)=4x'-~9f. 

4.  {5a+4b){oa-4b)=2Da'-lGb\ 

5.  [a:'-^b''){a'—b')=a'—b\ 

6.  {2am-^Sbn)  {2am—Sbn)=4a-'m-'—9¥n\ 

Art.  83. — If  we  divide  a^  by  a^,  since  the  rule  for  the  exponents 
requires  that  the  exponent  of  the  divisor  should  be  subtracted 

a^ 
from  that  of  the  dividend,  we  have  -.=a^-5_^-2^ 

a'' 
But,  since  the  value  of  a  fraction  is  not  altered  by  dividing  both 
terms  by  the  same  quantity,  (Art.  127),  if  we  divide  both  numer- 

ator  and  denominator  by  a^,  we  have  —^=-—. 
"^  a^     a^ 

Hence  a"'^=^^;,  since  each  equals  —y. 
a^  ^         a-" 

In  the  same  manner  by  subtracting  the  exponents 

a"* 

a" 

Or,  by  dividing  both  terms  by  a"*,   — -=-z-^ ; 
Hence, 0"*-"=^ .     Therefore, 

THEOREM   IV. 

The  reciprocal  of  a  qiianiiii/  is  equal  to  the  same  quantity  with  the 
sign  of  its  exponent  changed. 

Thus,  since  —  is  the  reciprocal  of  a"*  (Art.  51);  — =a— "*. 
a*"  a"* 

And  since is  the  reciprocal  of  a—"*;  =a"*. 

Also.    .    .    .  :=^=a&-"»;  — =a'"6-"; 

From  this  we  see,  that  anij  factor  may  be  transferred  from  one 
term  of  a  fraction  to  the  other,  if,  at  the  same  time,  the  sign  of  its 
exponent  be  changed. 
6 


66                     BAY'S   ALGEBRA,    PART   FIRST. 
Thu.: «=„i-:^l-_;=4_ 

Art.  §4. — Let  it  be  required  to  divide  a"^  by  a'\  By  the  rule 
for  the  exponents,  (Art.  73),  — =a'-^-''^=:a° ;  but  since  any  quantity 

is  contained  in  itself  once,  -;r=l. 

Similarly,  —=a"'-'^=a^ ;  but  --=1,  therefore  a'^^^l  since  each 

a"* 
is  equal  to  -— .     Hence, 

THEOREM    V. 

Ally  quantity  wJiose  exjjonent  is  0  is  equal  to  unity. 

This  notation  is  used,  when  we  wish  to  preserve  the  trace  of  a 
letter,  which  has  disappeared  in  the  operation  of  division.  Thus, 
if  it   is    required   to  divide   mhi^  by  mu'^  the    quotient  will   be 

-:=m^'''^n^~'^=mhi^^=m,  since  n^=l.     Now,  the  quotient  is  cor- 

mri 

rectly  expressed  either  by  inhi^,  or  m,  since  both  have  the  same 
value.  The  first  form  is  used,  when  it  is  necessary  to  show  that  oi 
originally  entered  as  a  factor  into  the  dividend  and  divisor. 

Art.  85. — 1.  If  we  divide  a^ — ¥  by  a — b,  the  quotient  will  be 
a+b. 

2.  If  we  divide  a^ — b^  by  a — b,  the  quotient  will  be  a^+«Z)+&l 

In  the  same  manner,  we  would  find,  by  trial,  that  the  difi'erence 
of  the  same  powers  of  two  quantities,  is  always  divisible  by  the 
dijfference  of  the  quantities.  The  direct  proof  of  this  theorem  is 
as  follows. 

Let  us  divide  a'^—b'^  by  a — b. 

a'^--b'^\a—b  

r, — -, a'^'^H — ^ 5 ^Quotient. 

=6(a"'~^ — 6"""^)    Remainder. 

In  performing  this  division,  we  see  that  the  first  term  of  the 
quotient  is  a"*~S  and  that  the  first  remainder  is  6(a"*~^ — &"*-^). 

The  remainder  consists  of  two  factors,  b  and  a"*""^ — Z>"^^.  Now, 
it  is  evident,  that  if  the  second  of  these  factors  is  divisible  by 
a — 6,  then  will  the  quantity  a"* — b'^  be  divisible  by  a — b.  Thus, 
if  a — b  is  contained  c  times  in  a'"~^ — b^~^,  the  whole  quotient  of 
a'" — 6*",  divided  by  a—b,  would  be  a"*-^-}-6c. 


ALGEBRAIC    THEOREMS.  G7 

From  this,  we  see  that  if  «"'-!— ^"^-i  jg  divisible  by  a~h,  then 
will  a'^—h"'  be  also  divisible  by  it.  That  is,  if  the  difference  of  the 
same  powers  of  two  quantities  is  divisible  by  the  difference  of  the 
quantities  themselves,  then  will  the  difference  of  the  next  higher  powers 
of  the  same  quantities,  be  divisible  by  the  difference  of  the  quantities. 
But  we  have  seen,  already,  that  a^ — b'^  is  divisible  by  a — b  ;  hence, 
it  follows,  that  a^ — W  is  also  divisible  by  a — b.  Then,  since  a^ — 6' 
is  divisible  by  a — b,  it  again  follows,  that  a* — b*  is  divisible  by  it ; 
and  so  on,  without  limit.     Hence,  we  have 

THEOREM  VI. 

The.  difference  of  the  same  poioers  of  two  quantities,  is  always 
divisible  by  the  difference  of  the  quantities. 

The  quotients  obtained  by  dividing  the  difference  of  the  same 
powers  of  two  quantities,  by  the  difference  of  those  quantities, 
follow  a  simple  law.     Thus: 

{a'—b'')-i-{a—b)=a+b. 

la^—li)^(^a—b)=a^+ab-{-b\ 

{a*—b')-^{a—b)=a^-i-a''b+ab'+b\ 

The  exponent  of  the  first  letter  decreases  by  unity,  while  that 
of  the  second  increases  by  unity. 

Art.  86. — Since  a"' — 6"*  is  always  divisible  by  a — b,  if  we  put 
— c  for  b,  then  a — b  will  become  a+c,  and,  since  6"*  Avill  become 
c*",  when  m  is  even,  as  2,  4,  6,  &c.,  and  — c"*,  when  m  is  odd,  as  3, 
5,  7,  &c.,  therefore,  a"* — ft"*  will  become  a'" — c"*,  when  m  is  even, 
and  a'"+c'",  when  m  is  odd,  because  a"'—b"'^=a"* — (— c"')=a*"-f  c™; 
therefore,  a"" — c"'  is  always  divisible  by  a-{-c,  when  m  is  even,  and 
^"•-f  c"*  is  always  divisible  by  a+c  when  m  is  odd.  These  truths 
are  expressed  in  the  following  theorems. 

THEOREM    VII. 

The  difference  of  the  even  poivers  of  the  same  degree  of  two  quan- 
tities, is  always  divisible  by  the  sum  of  the  quantities. 

Thus:   [a^-b-')^{a-\-h)=a-b. 

{a*~b*)-^{a+b)=a^~a^b+a¥—bK 

(a6_^6J  ^  la-{-b)=--a^—a*b-i-a^b''—a'b^+ab*—b\ 

R  E  V I  E  AV. — 80.  To  what  is  the  square  of  the  sum  of  two  quantities 
equal  ?  81.  To  what  is  the  square  of  the  difference  of  two  quantities  equal  ? 
82.  To  what  is  the  product  of  the  sum  and  difference  of  two  quantities 
equal  ?  83.  IIow  may  the  reciprocal  of  any  quantity  be  expressed  ?  How 
may  any  factor  be  transferred  from  one  term  of  a  fraction  to  the  other? 
In  what  other  form  may  a"'  be  written  ?  a-"'  ?  84.  What  is  the  value  of 
any  quantity  whose  exponent  is  zero  ? 


68  RAY^S   ALGEBRA,    PART    FIRST. 

THEOREM    VIII. 

The  sum  of  the  odd  powers  of  the  same  degree  of  two  quantities, 
is  always  divisible  by  the  sum  of  the  quantities. 
Thus :  {a^-\-b^) -^{a+b)=a^-ab-\-b\ 

\aP+b^)-^[a+b)=a^—a:%+a'b-'—ah^+b\ 
{a^-^b')-~{a+b)=za'^—a^b'{-an/—a''b''-^d'b^—ab^+b^ 


FACTORING. 

FACTORS,    AND    DIVISORS   OF    ALGEBRAIC    QUANTITIES. 

Art.  87. — A  divisor  or  measure  of  a  quantity,  is  any  quantity 
that  divides  it  without  a  remainder,  or  that  is  exactly  contained 
in  it.  Thus,  2  is  a  divisor  of  6 ;  and  a^  is  a  divisor  or  measure 
of  a^x. 

Art.  88. — A  prime  nuniber,  is  one  which  has  no  divisors  except 
itself  and  unity. 

A  composite  number,  is  one  which  has  one  or  more  divisors 
besides  itself  and  unity. 

Hence  all  numbers  are  either  prime  or  composite ;  and  every 
composite  number  is  the  product  of  two  or  more  prime  numbers. 

The  following  is  a  list  of  the  prime  numbers  under  100: 

1,  2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  37,  41,  43,  47,  53, 
59,  61,  67,  71,  73,  79,  83,  89,  97. 

The  composite  numbers  are,  4,  6,  8,  9,  10,  12,  &c. 
RULE, 
FOR    RESOLVING    ANY    COMPOSITE    NUMBER    INTO    ITS    PRIME    FACTORS. 

Divide  by  any  prime  number  that  will  exaetly  divide  it ;  divide  the 
quotient  again  in  the  same  manner;  and  so  continue  to  divide,  until 
a  quotient  is  obtained,  which  is  a  prime  number;  then,  the  last  quo- 
tient and  the  several  divisors,  loill  constitute  the  prime  factors  of  the 
given  number. 

R  E  sr  A  RK  . — The  reason  of  this  rule  is  evident,  from  the  nature  of  prime 
and  composite  numbers.  It  will  be  found  most  convenient  to  divide  first 
by  the  smallest  prime  number  that  is  a  factor. — See  Ray's  Arithmetic,  Part 
III.,  Factoring. 

R  E  VI E  w. — 85.  By  what  is  the  difference  of  the  same  powers  of  two  quan- 
tities always  divisible?  86.  By  what  is  the  difference  of  the  even  powers 
of  the  same  degree  of  two  quantities  always  divisible?  By  what  is  the 
sum  of  the  odd  powers  of  the  same  degree  of  two  quantities  always 
divisible  ? 


FACTORING.  6'J 


EXAxMPLES. 

1.  The  composite  numbers  under  100,  that  is,  4,  6,  8,  &c., 
may  be  given  as  examples.  Every  pupil  should  learn  to  give  the 
factors  of  these  quantities  readily. 

2.  What  are  the  prime  factors  of  105? Ans.  3,  5,  7. 

3.  What  are  the  prime  factors  of  210?  .    .    .      Ans.  2,  3,  5,  7. 

4.  Resolve  4290  into  its  prime  factors.  .    Ans.  2,  3,  5,  11,  13. 

Art.  89. — A  prime  quantity,  in  Algebra,  is  one  which  is  exactly 
divisible  only  by  itself  and  by  unity.  Thus,  a,  b,  and  6+c  are 
prime  quantities;  while  ab  and  ab-j-ac  are  not  prime. 

Art.  90. — Two  quantities,  like  two  numbers,  are  said  to  be 
prime  to  each  other,  or  relatively  prime,  Avhen  no  quantity  except 
unity  will  exactly  divide  them  both.  Thus,  ab  and  cd  are  prime 
to  each  other. 

Art.  91. — A  composite  number,  or  a  composite  quantity,  is  one 
which  i^  the  product  of  two  or  more  factors,  neither  of  which  is 
unity.  Thus,  ax  is  a  composite  quantity,  of  which  the  factors  are 
a  and  x. 

Remark. —  A  monomial  may  be  a  composite  quantity,  as  ax;  and  a 
polynomial  may  not  be  a  composite  quantity,  as  a^-Yx'^. 

Art.  92. — To  separate  a  monomial  into  its  prime  factors. 

RULE. 

Resolve  the  coefficient  into  its  prime  factors;  then  these,  ivith  the 
literal  factors  of  the  monomials,  will  form  the  prime  factors  of  the 
given  quantity.     The  reason  of  this  rule  is  self-evident. 

Find  the  prime  factors  of  the  following  nominals: 

1.  l^d^bc Ans.  3X5. a. a.6.c. 

2.  2lal)'d Ans.  SX7M.b.b.d. 

3.  S^abc^x Ans.  bX7.a.b.c.c.x. 

4.  SQa-m'^n Ans.  3Xl3.a.a.m.m.7j. 

Art.  93. — To  separate  a  polynomial  into  its  factors,  when  one 
of  them  is  a  monomial  and  the  other  a  polynomial. 

Review. — 87.  What  is  the  divisor  of  a  quantity?  88.  Whal  is  a  prime 
number  ?  What  is  a  composite  number  ?  Name  several  of  the  prime  num- 
bers, beginning  with  unity.  Name  several  of  the  composite  numbers, 
beginning  with  4.  What  is  the  rule  for  resolving  any  composite  number 
into  its  prime  factors?  89.  What  is  a  prime  quantity?  Give  an  example. 
yO.  When  are  two  quantities  prime  to  each  other?  Give  an  example.  91. 
What  is  a  composite  quantity  ?  Give  an  example.  92.  What  is  the  rule 
for  separating  a  monomial  into  its  prime  factors? 


70  RAY'S    ALGEBRA,    PART    FIRST. 

RULE. 

Divide  the  given  quaniity  by  the  greatest  monomial  that  loill  exactly 
divide  each  of  its  terms.  Then  the  monomial  divisor  will  be  one  fac- 
tor, and  the  quotient  the  other.  The  reason  of  this  rule  is  self- 
evident. 

Separate  the  following  expressions  into  factors ; 

1.  x-\-ax Ans.  x{l-\-a). 

2.  am+ac Ans.  a(m+c) 

3.  bc'-^bcd Ans.  bc[c+cl) 

4.  Ax'+Qxy Ans.  2x[2x+^y) 

5.  Qax^y-^r^bxif — \2cxhj Ans.  3xy(2ax'+36y — 4.cx) 

6.  bax- — 'dbax?y-[-bd^x^y Ans.  5«x'^(l — 7xy-^axy) 

7.  Ua^x-y +21  d'xY—S5a^xy\    .    Ans.  7a^xy{2ax+Sxy—5ay) 

8.  66c-'x--15&c-''— 36V Ans.  3bc'{2x-5c-b) 

9.  a^cm'^-\-d^c'^m'^ — a'^cni^ Ans.  a-cm'^{a-'rc — m) 

Art.  94. — To  separate  a  quantity  which  is  the  product  of  two 

or  more  polynomials,  into  its  prime  factors. 

No  genei*al  rule  can  be  given,  for  this  case.  When  the  given  quantity 
does  not  consist  of  more  than  three  terms,  the  pupil  will  generally  be  able 
to  accomplish  it,  if  he  is  familiar  with  the  theorems  in  the  pi'eceding  section. 

1st.  Any  trinomial  can  be  separated  into  two  binomial  factors, 
when  the  extremes  are  squares  and  positive,  and  the  middle  term 
is  twice  the  product  of  tlie  square  roots  of  the  extremes.  See 
Articles  79  and  80. 

Thus:  d'+2ab-}-b'={a+b)[a+b). 
d'—2ab+¥={a-b){a—b). 

2d.  Any  binomial,  which  is  the  difference  of  two  squares,  can 
"be  separated  into  two  f^ictors,  one  of  which  is  the  sum,  and  th(3 
other  the  difference  of  the  roots.     See  Art.  81. 

Thus:  d'—b'^{a-i'b){a—b). 

3d.  When  any  expression  consists  of  the  difference  of  the 
same  powers  of  two  quantities,  it  can  be  separated  into  at  least 
two  Victors,  one  of  which  is  the  difference  of  the  quantities.  See 
Art.  84. 

Thus:  a'"— 6'"=(a-6)(a'"-^+a'"-26 +a5'"-^+Zy"'-^), 

where  a,  b,  and  m,  may  be  any  quantities  whatever. 

In  this  case,  one  of  the  factors  being  the  difference  of  the  quan- 
tities, the  other  will  be  found  by  dividing  the  given  expression  by 
this  difference.  Thus,  to  find  the  other  factor  of  a^ — b^,  divide  by 
a — b,  the  quotient  will  be  found  to  be  a'^-j-ab-'rb^ ;  hence,  a^ — b^ 


FACTORING. 


71 


In  a  similar  manner,  a^—lr'=[a—b)[a^-\-a^h-\-d'h'^^al/-^¥). 

4th.  When  any  expression  consists  of  the  difference  of  the  even 
powers  of  two  quantities,  higher  than  the  second  degree,  it  can 
be  separated  into  at  least  three  factors,  one  of  which  is  the  sum, 
and  another  the  difference  of  the  quantities.  See  Articles  85 
and  86. 

Thus,  a*— 6*  is  exactly  divisible  by  a-\-l>,  according  to  Article 
66  ;  and,  according  to  Article  85,  it  is  exactly  divisible  by  a — 6  ; 
hence,  it  is  exactly  divisible  by  both  a-{-h  and  a — 6;  and  the  other 
factor  will  be  found  by  dividing  by  their  product.  Or,  it  may  be 
separated  into  factors,  according  to  paragraph  2d,  above,  thus: 

5th.  When  any  expression  consists  of  the  sum  of  the  odd  pow- 
ers of  two  quantities,  it  may  be  separated  into  at  least  two  factors, 
one  of  which  is  the  sum  of  the  quantities  (See  Art.  86).  The 
other  factor  will  be  found,  by  dividing  the  given  expression  by 
this  sum.  Thus,  we  know  that  a^-\-h^,  is  exactly  divisible  by  a+6, 
and  by  division,  we  find  the  other  factor  to  be  a^ — ah-\-h'^ ;  hence, 

Separate  the  following  expressions  into  their  simplest  factors. 


1.  x-^+2xy+yl 

2.  9ci'-\-\2ah-\-4.h\ 

3.  4+12a;+9a;2. 

4.  Tn^—'iZmn-^n'^. 

5.  a?—2abx+h''x''. 

6.  4a;2— 20a;z+2522. 


7. 


-r 


8.  9m^—\Qn\ 


1.  [x^y){xMj)' 

2.  (3a4-26)(3a+26). 

3.  (2+3x)(2+3x). 

4.  {m—n){m — n). 

5.  {a—bx){a—bx). 


2). 


9.  aW-c'd'\ 

10.  a''x-x\ 

11.  x*-b\ 

12.  tf+l. 

13.  a;^-l. 

14.  8a'-21b\ 

15.  a^-}-b\ 

16.  a'-b\ 

ANSWERS. 

10.  x{a-rx){a—x). 

1 1 .  (x-2+6-^)  [x'  —  b')  =  {x''-\-b'') 
{x-}-b){x-b). 

12.  {y+lW-!/-{-l). 

13.  (x-l)(x2+x+l). 

14.  {2a—3b){4a'-^6ab+9b'-). 

15.  [a^b)  {a'—a^b-haW-ab^ 


6.  (2a:-52)(2x 

7.  (x+y)(x— y). 

8.  {Sm+4n){Sm—4n) 

9.  {ab+cd){ab—cd). 
16.  {a'-^b'){a'-b^)=:{a'+b'){a-b){d'+ab-\-b'). 

={a+b){a'-ab-^b'){a-b){d'+ab+b'). 
={a-^b)  [a—b)  {d'—ab-^b')  (a^+aft+ft^). 


72  RAY'S    ALGEBRA,    PART    FIRST. 

Art.  95. — To  separate  a  quadratic  trinomial  into  its  factors. 

A  quadratic  trinomial  is  of  the  form,  a;^+ax+^,  in  which  the 
signs  of  the  second  and  third  terms  may  be  either  plus  or  minus. 
"When  this  operation  is  practicable,  the  method  of  doing  it,  may- 
be learned  by  observing  the  relation  that  exists  between  two  bino- 
mial factors  and  their  product. 

1.  (x4-«)(a;+6)=x--+(a+6)aj+a6. 

2.  {x — a){x — b)=x'^ — {a-\-b)x-\-ab. 

3.  {x-{-a){x — 6)=x^+(a — b)x — ab. 

4.  {x — a){x-^b)^=^x'^-\-{b — a)x — ab. 

From  the  preceding,  we  see,  that  when  the  first  term  of  a  quad- 
ratic trinomial  is  a  square,  with  the  coefficient  of  its  second  term 
equal  to  the  sum  of  any  tAvo  quantities,  which,  being  multiplied 
together,  will  produce  the  third  term,  it  may  be  resolved  into  two 
binomial  factors  by  inspection. 

Decompose  each  of  the  following  trinomials  into  two  binomial 
factors. 

1.  x2+5x+6 Ans.  (x+2)(a:+3). 

2.  a24_7«,_|-i2 Ans.  (a+3)(a+4). 

3.  a:2-5x-+6 Ans.  (x— 2)(a:-3). 

4.  x^— 9x+20 Ans.  (x— 4)(x— 5). 

5.  x2+x-6 Ans.  (x+3)(x-2). 

6.  x2-x— 6 Ans'.  (x— 3)(xH-2). 

7.  x^+x— 2 Ans.  (x+2)(x-l). 

8.  x^— 13x+40 ,  Ans.  (x-8)(x-5). 

9.  x^— 7x-8 Ans.  (x-8)(x+l). 

10.  x'^+7x— 18 Ans.  (x+9)(x-2). 

11.  x^— x-30 .  Ans.  (x— 6)(x+5). 

In  the  same  manner,  we  may  often  separate  other  trinomials 
into  factors,  by  first  taking  out  the  monomial  factor  common  to 
each  term. 

Thus,  5ax2-10ax— 40a=5a(x2-2x— 8)=5a(x-4)(x+2). 

12.  3x^+1 2x- 15 Ans.  3(x-f5)(x-l). 

13.  aV— 9a'^x+14a2 Ans.  a''{x—7){x—2). 

14.  2abx''—Uabx—60ab Ans.  2«5(x-l())(x+3). 

15.  2x3— 4x2— 30x Ans.  2x(x---5){x-f3). 

Review. — 93.  What  is  tho  rule  for  separating  a  polynomial  into  its 
prime  factors,  when  one  of  them  is  a  monomial,  and  the  other  a  polyno- 
mial ?  94.  AVhen  can  a  trinomial  be  separated  into  two  binomial  factors  "^ 
What  are  the  factors  of  m^^+2mn-j-n^?  Of  c^—2cd-\-d^?  When  can  a  bi- 
uomial  be  separated  into  two  binomial  factors  ?  What  are  the  factors  of 
x^—y-i  ?  Of  9a2— 16^2  ?  What  is  one  of  tho  factorb  of  a^—b'^  ?  Of  a"^~b3  -> 
Of  x*—y*  ?     What  are  two  of  rho  factors  of  fi*—h*?     Of  n^—h^? 


GREATEST   COMMON   DIVISOR.  73 

Art.  96.  —  The  principal  use  of  factoring,  is  to  shorten  the 
work,  and  simplify  the  results  of  algebraic  operations.  Thus, 
when  it  is  required  to  multiply  and  divide  by  algebraic  express- 
ions, if  the  multiplier  and  divisor  contain  a  common  factor,  it  may 
be  canceled,  or  left  out  in  both,  without  affecting  the  value  of  the 
result.  Thus,  if  it  is  required  to  multiply  any  quantity  by  a?' — 6^ 
and  then  to  divide  the  product  by  a-^h,  the  result  will  be  the  same 
as  to  multiply  at  once  by  a—b. 

Whenever  there  is  an  opportunity  of  canceling  common  factors, 
the  operations  to  be  performed  should  be  merely  indicated,  as  the 
common  factors  will  then  be  more  easily  discovered.  The  pupil 
will  see  the  application  of  this  principle,  by  solving  the  following 
examples. 

1.  Multiply  a—h  by  x^-\-2xy+if,  and  divide  the  product  by  ic+y. 

x+y  x+y  ^         '^  ~^^' 

=^ax-\-ay — hx — hy. 

2.  Multiply  X — 3  by  x^ — 1,  and  divide  the  product  by  x~\,  by 
factoring.  Ans.  a;^— 2a:— 3. 

3.  Divide  2^+1  by  2+1,  and  multiply  the  quotient  by  z^ — 1,  by 
factoring.  Ans.  z*—z^-\-z—\. 

4.  Divide  Qa^c — 12a6c4-66^c  by  2ac — 26c,  by  factoring. 

Ans.  3  (a — 6). 

5.  Multiply  6ax-+9ay  by  4x^ — 9?/^  and  divide  the  product  by 
4a;2+12a:y+9^/^  by  factoring.  Ans.  3«(2x— 3?/). 

6.  Multiply  x^—5a:+ 6  by  a;'^—7x+ 12,  and  divide  the  quotient 
by  a;2—6x+ 9,  by  factoring.  Ans.  (x— 2) (a;— 4). 

Other  examples  in  which  the  principle  may  be  applied,  will  be 
found  in  the  multiplication  and  division  of  fractions. 


GREATEST    COMMON    DIVISOR. 

Art.  97. — Any  quantity  that  will  exactly  divide  two  or  more 
quantities,  is  called  a  common  divisor,  or  common  measure,  of  those 
quantities.  Thus,  2  is  a  common  divisor  of  8  and  12;  and  a  is 
a  common  divisor  of  ah  and  a'^x. 

Remark. — Two  quantities  may  sometimes  have  more  than  one  com- 
mon divisor.     Thus,  8  and  12  havo  two  common  divisors,  2  and  4. 

Review.— 94.  What  is  one  of  the  factors  of  a^-\-h^'i    What  is  one  of 
the  factors  of  .v>+y^l     95.  What  is  a  quadratic  trinomiiil  ? 


74  RAY'S   ALGEBRA,    PART   FIRST. 

Art.  98. — That  common  divisor  of  two  quantities,  which  is  the 
greatest,  both  with  regard  to  the  coefficients  and  exponents,  is 
called  their  greatest  common  divisor,  or  greatest  common  measure. 
Thus,  the  greatest  common  divisor  of  Aa^xy  and  Qa^xh/  is  2a'^xij. 

Art.  99o — Quantities  that  have  a  common  divisor,  are  said  to 
be  commensurable;  and  those  that  have  no  common  divisor,  are 
said  to  be  incommensurable.  Incommensurable  quantities  are  also 
said  to  be  prime  to  each  other,  or  relatively  prime. 

Art.  100. — To  find  the  greatest  common  divisor  of  two  or  more 
monomials. 

1.  Let  it  be  required  to  find  the  greatest  common  divisor  of  the 
two  monomials.  Gab  and  15a'^c. 

By  separating  each  quantity  into  its  prime  factors,  we  have 
(yab=2XSab,  15a^c=SXbaac. 

Here  Ave  see,  that  3  and  a  are  the  only  factors  common  to  both 
terms;  hence,  both  the  quantities  can  be  exactly  divided,  either 
by  3  or  a,  or  by  their  product  3a,  and  by  no  other  quantity 
whatever;  consequently,  3a  is  their  ;  reatest  common  divisor. 
Hence,  the 

RULE, 

FOR    FINDING     THE     GREATEST     COMMON     DIVISOR     OF     TWO     OR     MORE 
MONOMIALS. 

Resolve  the  quantities  into  their  prime  factors  ;  then,  the  product 
of  those  factors  that  are  common  to  each  of  the  terms,  will  form  the 
greatest  common  divisor. 

Note. — The  greatest  common  divisor  of  the  literal  parts  of  the  quan- 
tities, may  generally  be  more  easily  found  by  inspection,  by  taking  each 
letter  with  the  highest  power,  that  is  common  to  all  the  quantities. 

2.  Find  the  greatest  common  divisor  of  4a^a:^,  Ga^x^,  and  10a*x. 
4aV=2X2a'^x^  Here  we  see,  that  2,  a^  and  x  are  the  only 
6tt^x^=2X3a^a;^     factors  common  to  all  the  quantities ;  hence, 

10a*x=2X5tt*x      2a^x  is  the  greatest  common  divisor. 
Find  the  greatest  common  divisor  of  the  following  quantities. 

3.  4aV,  and  lOax^ Ans.  2aa;^ 

4.  9ahc^  and  I2bc*x Ans.  36c^ 

5.  4a^b'^x^g^,  and  Sa^xi^g^ Ans.  \(]?xhf. 

6.  3aV'.  6aV/A  and  9ahfz Ans.  3ay. 

7.  8axy^^  ^ix^z^,  and  24aV22 Ans.  4xV. 

8.  GaW^  l2aYz\  9aV/,  and  24aYz Ans.  3ay. 

Art.  101. — To  find  the  greatest  common  divisor  of  two  poly- 
nomials. 

First.  Let  AD  and  BD  be  either  two  monomials,  or  polynomials, 
of  which  D  is  a  common  divisor ;  and  let  AD  be  greater  than  BD. 


GREATEST   COMMON   DIVISOR.  75 

Divide  AD  by  BD ;  then,  if  it  gives  an  exact  quotient,  BD  must 
be  the  greatest  common  divisor,  since  no     fiT)\  atj^O 
quantity  can  have  a   divisor  greater  than  'RDO 

itself.     But,  if  BD  is  not  contained  an  exact 

number  of  times  in  AD,  suppose  it  is  con-  ^         ^^H    ^ 

tained  Q  times  with  a  remainder,  which  may  be  called  R.  Then, 
since  the  remainder  is  found,  by  subtracting  the  product  of  the 
divisor  by  the  quotient,  from  the  dividend,  we  have  R=AD — BDQ. 

Dividing  both  sides  by  D,  we  get  — =A— BQ ;  But  A  and  BQ  are 

R        .      . 

each  entire  quantities;  hence  ^p-,  which  is  equal  to  their  difference, 

must  be  an  entire  quantity.  Hence,  it  follows,  that  ant/  common 
divisor  of  two  quantities,  will  always  exactly  divide  their  remainder 
after  division.  And,  since  the  greatest  common  divisor  is  a  com- 
mon divisor,  it  follows  that  the  greatest  common  divisor  of  two 
quantities,  will  always  exactly  divide  their  remainder  after  division. 

Remark  . — In  the  above  article,  we  have  used  two  axioms,  which  may 
be  new  to  some  pupils.  They  are,  first:  If  two  equal  quantities  be  divided 
hy  the  same  quantity,  their  quotients  loill  he  equal.  And,  second:  The 
difference  of  two  entire  quantities  is  also  an  entire  quantity.  The  pupil  can 
easily  see,  that  the  sum,  or  difference  of  two  whole  numbers  must  also  be 
a  whole  number ;  and,  that  the  same  is  likewise  true  of  two  entire  quan- 
tities. This,  and  the  next  article  will  both  be  better  understood  by  the 
pupil,  after  he  has  studied  simple  equations. 

Art.  102. — Second.  Suppose,  now,  that  it  is  required  to  find 
the  greatest  common  divisor  of  two  polynomials,  A  and  B,  of 
which  A  is  the  greater. 

If  we  divide  A  by  B,  and  there  is  no     b\  a /n 

remainder,  B  is,  evidently,  the  greatest     gg 

common  divisor,  since  it  can  have  no  di-     — — — :      „    ,      ,, 

,      ,,       .,    ,p  A— BQ=R,  1st  Rem. 

visor  greater  than  itself. 

Dividing  A  by  B,  and  calling  the  quo-  R)B(Q' 

tient  Q^  if  there  is  a  remainder  R,  it  is  j^q/ 

evidently  less  than  either  of  the  quanti-  „ RO'—R'  2d  Rem 

ties  A  and  B  ;  and,  by  the  preceding  the-  ' 

orem,  it  is  also  exactly  divisible  by  the  A=BQ+R       Since  the 

greatest  common  divisor;  hence,  the  great-  B==RQ'+R'  dividend  is 
est  common  divisor  must  divide  A,  B,  and  equal  to  the 

R,  and  can  not  be  greater  than  R.     But  product  of  the  divisor  by 

if  R  will  exactly  divide  B,  it  will  also  ex-  ^^e  quotient,  plus  the  re- 

nctly  divide  A,  since  A::^BQ+R,  and  will  ™^"^ 
be  the  greatest  common  divisor  sought. 


7G  RAY'S   ALGEBRA,    PART    FIRST. 


Suppose,  however,  that  when  we  divide  R  into  B,  to  ascertain 
if  it  will  exactly  divide  it,  we  find  that  the  quotient  is  Q',  with  a 
remainder,  R'.  Now,  it  has  been  shown,  that  whatever  exactly 
divides  two  quantities,  will  divide  their  remainder  after  division ; 
then,  since  the  greatest  common  divisor  of  A  and  B,  has  been 
shown  to  divide  B  and  R,  it  will  also  divide  their  remainder  R', 
and  can  not  be  greater  than  R^  And,  if  R'  exactly  divides  R,  it 
will  also  divide  B,  since  B^RQ'+B' ;  and  whatever  exactly  divides 
B  and  R,  will  also  exactly  divide  A,  since  A==BQ-f  R ;  therefore, 
if  R'  exactly  divides  R,  it  will  exactly  divide  both  A  and  B,  and 
will  be  their  greatest  common  divisor. 

In  the  same  manner,  by  continuing  to  divide  the  last  divisor  by 
the  last  remainder,  it  may  always  be  shown,  that  the  greatest  com- 
mon divisor  of  A  and  B  will  exactly  divide  every  new  remainder, 
and,  of  course,  can  not  be  greater  than  either  of  them.  It  may, 
also,  always  be  shown,  as  above,  in  the  case  of  R',  that  any 
remainder,  which  exactly  divides  the  preceding  divisor,  will  also 
exactly  divide  A  and  B.  Then,  since  the  greatest  common  divisor 
of  A  and  B  can  not  be  greater  than  this  remainder,  and,  as  this 
remainder  is  a  common  divisor  of  A  and  B,  it  will  be  their  great- 
est common  divisor  sought. 

To  illustrate  the  same  principle  by  numbers,  let  it  be  required 
to  find  the  greatest  common  divisor  of  14  and  20. 

If  we  divide  20  by  14,  and  there  is  no  remain-  14)20(1 
der,  14  is,  evidently,  the  greatest  common  divisor,  14 

since  it  can  have  no  divisor  greater  than  itself.  6)14(2 

Dividing  20  by  14,  we  find  the  quotient  is  1,  and  X2 

the  remainder  6,  which  is,  necessarily,  less  than  ~2)C('^ 

either  of  the  quantities,  20  and  14;  and  by  the  n 

theorem,  Article  98,  it  is  exactly  divisible  by  their  — 

greatest  common  divisor ;  hence,  the  greatest  common  divisor 
must  divide  20,  14,  and  6,  and  cannot  be  greater  than  6.  Now, 
ifGvvill  exactly  divide  14,  it  will  also  exactly  divide  20,  since 
20=14-f6,  and  will  be  the  greatest  common  divisor  sought. 

But  when  we  divide  6  into  14,  to  ascertain  if  it  will  exactly 
divide  it,  we  find  that  the  quotient  is  2,  Muth  a  remainder,  2 ;  then, 

Review. — 95.  When  can  a  quadratic  trinomial  be  separated  into  bino- 
mial factors  ?  96.  What  is  the  principal  use  of  factoring  ?  97.  What  is  a 
common  divisor  of  two  or  more  quantities  ?  Give  an  example.  98.  What 
is  the  greatest  common  divisor  of  two  quantities  ?  Give  an  example.  99. 
When  arc  quantities  commensurable  ?  When  are  quantities  incommen- 
surable ?  100.  How  do  you  find  the  greatest  common  divisor  of  two  or 
more  monomials  ?  101.  Prove  that  any  common  divisor  of  two  quantities 
will  always  exactly  divide  their  remainder,  after  division. 


GREATEST   COMMON   DIVISOR.  77 

by  the  preceding  theorem,  the  greatest  common  divisor  of  14  and 
6  Avill  also  divide  2,  and  therefore,  can  not  be  greater  than  2. 
Now,  if  2  will  exactly  divide  6,  it  will,  also,  exactly  divide  14, 
since  14=6X2+2;  and  whatever  will  exactly  divide  6  and  14, 
will  also  divide  20.  But  2  exactly  divides  6 ;  hence  it  is  the 
greatest  common  divisor  of  14  and  20. 

Art.  103* — When  the  remainders  decrease  to  unity,  or  when 
we  arrive  at  a  remainder  which  does  not  contain  the  letter  of 
arrangement,  we  conclude  that  there  is  no  common  divisor  to  the 
quantities. 

Art.  104. — If  one  of  the  quantities  contains  a  factor  not  found 
in  the  other,  it  may  be  canceled  without. affecting  the  common 
divisor  (see  example  3);  and  if  both  quantities  contain  a  common 
factor,  it  may  be  set  aside  as  a  factor  of  the  common  divisor ;  and 
we  may  proceed  to  find  the  greatest  common  divisor  of  the  other 
factors  of  the  given  quantities.  This  is  self-evident.  See  Ex- 
ample 2. 

Art.  105* — We  may  multiply  either  quantity,  by  a  factor  not 

found  in  the  other,  without  affecting  the  greatest  common  divisor. 

2abx 
Thus,  in  the  fraction  q"/"*  ^^^  greatest  common  divisor  of  the 

two  terms,  is  evidently  ab.  Here,  we  may  cancel  the  factors  2 
and  X  in  the  numerator,  or  3  and  c  in  the  denominator,  without 

affecting  the  common  divisor ;  for  the  common  divisor  of  o~r"»  ^^ 

„  2ahx  .      , ...    , 
01  — ;— ,  IS  still  ab. 
ab 

If  we  multiply  the  dividend  by  4,  a  factor  not  found  in  the  divi- 
sor, we  have  ^   ,  ,  of  which  the  common  divisor  is  still  ab. 
Sabc 

In  the  same  manner  we  may  multiply  the  divisor  by  any  factor 
not  found  in  the  dividend,  and  the  common  divisor  Avill  still  remain 
the  same. 

If,  however,  we  multiply  the  numerator  by  3,  which  is  a  factor 

of  the  denominator,  the  result  is  o-^,  of  which  the  greatest  com- 

oaoc 

mon  divisor  is  Sab,  and  not  ab  as  before.     Hence,  we  see,  that  the 

greatest  common  divisor  will  be  changed,  by  multiplying  one  of 

the  quantities  by  a  factor  of  the  other. 

Review. — 102.  Show,  that  by  dividing  the  last  divisor  by  the  last 
remainder,  the  greatest  common  divisor  of  two  polynomials  will  exactly 
divide  both  the  first  and  second  remainders  after  division. 


78  RAY'S   ALGEBRA,    PART   FIRST. 

Art.  106. — In  the  general  demonstration,  Art.  101,  it  has  been 
shown,  that  the  greatest  common  divisor  of  two  quantities,  also 
exactly  divides  each  of  the  successive  remainders  ;  hence,  the  pre- 
ceding principles  apply  to  the  successive  remainders  that  arise,  in 
the  course  of  the  operations  necessary  to  find  the  greatest  common 
divisor. 

The  preceding  principles  will  be  illustrated  by  some  examples. 

1.  Find  the  greatest  common  divisor  of  r^ — x^  and  x* — x^y^. 

Here  the  second  quantity  contains  x^  as  a  factor,  but  it  is  not  a 
factor  of  the  first;  we  may,  therefore,  cancel  it,  and  the  second 
quantity  becomes  x'^—y^.     Divide  the  first  by  it. 

After  dividing,  we  find  that  y"^  is  a  factor  of  the  ^ 3     |™2 «,2 

remainder,  but  not  of  x'^—y"^,  the  dividend.     Hence,  3         ^ ■—■ 

by  canceling  it,  the  divisor  becomes  x — y  ;  then,  di-  d. V'*' 

viding  by  this,  wo  find  there  is  no  remainder;  there-  xy'^ — y^ 

fore  X — y  is  the  greatest  common  divisor.  or,   (x — y)y'^ 

x^—7f     \x—y 


x^—xy         [x+y 
xy—if 

2.  Find  the  greatest  common  divisor  of  oi^^w^a?  and  a;* — a^x^. 


r'+a' 


The  factor  a;2  is  common  to  both  these  quantities  ; 

it  therefore  forms  part  of  the  greatest  common  divi-  . 

6or,  and  may  be  taken  out  and  reserved.     Doing  a.      a  x          [x 

this,    the   quantities   become   jc^^a^x   and   x^ — cf2.  d^x-\-d? 

The  first  quantity  still  contains  a  common  factor,  x,  or,  (x+ala^ 

which  the  latter  docs  not;    canceling  this,  it  be-     ;^ ^2      \x-\-a 

comes  a'3_|_„3.      Then,   proceeding   as    in   the   first  x'^-Vax    ix—a 

example,  we  find  the  greatest  common  divisor  is  — — r- 

tc\x^a),  —ax— a' 


3.  Find  the  greatest  common  divisor  of  5a^+10a*x+5aV  and 

a3x+2aV+2ar'+a;*. 

Here  5«3  is  a  factor  of    the  first     a'J^^cC-X^^ax^^^\a^-\-^ax^x^ 

quantity  only,  and  x,  of  the  second        3     ^^   2     I       2  i — — 

only.     Suppressing  these  factors,  and     ^  +^«  X-Yax  [a 

proceeding  as  in  the  previous  exam-  ax^+ar* 

pies,   we  find  a-l^x   is    the   greatest  or,  {a-\-x)x'^ 

common  divisor. 

a'^-\-2ax-\-x^     \a-\-x 
a^-j-ax  {a-\-x 


GREATEST    COMMON   DIVISOR. 


79 


2a* 


'—Gx' 


4.    Find  the  greatest   common  divisor  of   2a*— aV— 6a;*  and 

4aH-6aV— 2aV— Sa:^. 

In  solving  this  example,  4a^-\-Qa''x'—2a'^a^—3x^     |2a*— aV— 6x* 

there  are  two  instances  in  4a'^2a'^'-l2ax*  (Ta " 

wiiich  it  IS   necessary  to  — i- 

multiply  the   dividend,  in               8aV— 2aV+12aa:*— Sa;^ 
order  that  the  coefficient  of       or,  {8a^—2a^x-\-  I2ax'^ — Sar^jx^ 
the  first  term  may  be  ex- 
actly divisible  by  the  di- 
visor.   See  Art.  105.    The  4 

greatest  common  divisor     8a'-4d'x^~24x' \8a^-2a\i^A2ax'-3x^ 

IS  found  to  be  2a''^4-3a;!^.  qa     ct  ^     ,  m  •,   , — n — , ; 

^  gg*— 2a^a;+ 1 2a  V— 3qa:^  (a 

2a^x—  1 6a^x-^ +3ax=^— 24a;* 
4 

Sa^a;— 64aV+ 1 2aar^— 96a;*(a: 
8a^a;-~  2aVM-12ar^-  Sx* 

—G2aV 93a;* 

or,  — 3Ia;'-^(2a24-3a;''^) 

8a3_2a2a:-f  12aa;2— 3x3|2aH:3^ 
8a3  -fl2ax2  ^4^^^. 


-2a2a;- 
-2a^a;- 


-3a;' 
-3a;» 


From  the  preceding  demonstrations  and  examples,  we  derive  the 

RULB, 

FOR   FINDING    THE    GREATEST   COMMON    DIVISOR   OF   TWO   POLYNOMIALS. 

1  St.  Divide  the  greater  pohjnomial  by  the  less,  and  if  there  is  no 
remainder,  the  less  quantity  will  be  the  divisor  sought. 

2d.  If  there  is  a  remainder,  divide  the  first  divisor  by  it,  and  con- 
tinue to  divide  the  last  divisor  by  the  last  remainder,  until  a  divisor 
is  obtained,  which  leaves  no  remainder;  this  will  be  the  greatest  com- 
mon divisor  of  the  two  given  polynomials. 

Remarks. — 102.  Explain  the  principles  used,  in  finding  the  greatest 
common  divisor,  by  finding  it  for  the  numbers  14  and  20.  103.  When  do 
we  conclude  that  there  is  no  common  divisor  to  two  quantities  ?  104.  How 
is  the  comnion  divisor  of  two  quantities  affected,  by  canceling  a  factor  in 
one  of  them,  not  found  in  the  other?  When  both  quantities  contain  a  com- 
mon factor,  how  may  it  be  treated  ?  105.  How  is  the  greatest  common 
divisor  of  two  quantities  affected,  by  multiplying  either  of  them  by  a  factor 
not  found  in  the  other?  What  is  the  rule  for  finding  the  greatest  common 
divisor  of  two  polynomials  ?  How  do  you  find  the  greatest  common  divisor 
of  three  or  more  quantities  ? 


80  RAY'S   ALGEBRA,    PART    FIRST. 

Notes. — 1.  When  the  highest  power  of  the  leading  letter  is  the  same 
in  both,  it  is  immaterial  which  of  the  quantities  is  made  the  dividend. 

2.  If  both  quantities  contain  a  common  factor,  let  it  be  set  aside,  as  form- 
ing a  factor  of  the  common  divisor,  and  proceed  to  find  the  greatest  com- 
mon divisor  of  the  remaining  factors,  as  in  Example  2. 

3.  If  either  quantity  contains  a  factor  not  found  in  the  other,  it  may  be 
canceled,  before  commencing  the  opei-ation,  as  in  Example  3.     See  Art.  104. 

4.  Whenever  it  becomes  necessary,  the  dividend  may  be  multiplied  by 
any  quantity  which  will  render  the  first  term  exactly  divisible  by  the  divi- 
sor.    See  Art.  105. 

6.  If,  in  any  case,  the  remainder  does  not  contain  the  leading  letter,  that 
is,  if  it  is  independent  of  that  letter,  there  is  no  common  divisor. 

6.  To  find  the  greatest  common  divisor  of  three  or  more  quantities,  first 
find  the  greatest  common  divisor  of  two  of  them;  then,  of  that  divisor  and 
one  of  the  other  quantities,  and  so  on.  The  last  divisor  thus  found,  will  be 
the  greatest  common  divisor  sought. 

7.  Since  the  greatest  common  divisor  of  two  or  more  quantities  contains 
all  the  factoi-s  common  to  these  quantities,  it  may  be  found  most  easily  by 
separating  the  quantities  into  factors,  where  this  can  bo  done,  by  means  of 
the  rules  in  the  preceding  article. 

Find  the  greatest  common  divisor  of  the  following  quantities. 

5.  5a^-\-5ax  and  a^ — x^ Ans.  a-\-x. 

6.  x^ — a^x  and  a^ — a' Ans.  x — a. 

7.  x' — c'^x  and  x^-\-2cx-\-c^ Ans.  x-\-c. 

8.  x''+2x—3  and  a;2+5x+6 Ans.  x+S. 

9.  6a'+Uax-}-Sx' and  Ga'+Jax—Sx'.     .    .    .     Ans.  2a4-3a:. 

10.  a* — x*  and  a^-\-a^x — ax^ — x^ Ans.  a^—x'K 

11.  a^ — 5ax-{-4x^  and  a^ — a^x-\-Sax^ — 3a^ Ans.  a — x. 

12.  a^x^~dh/  and  x'-^^^y^ Ans.  x'^+y^. 

13.  a^— x*  and  d}^ — x^' Ans.  a — x. 


LEAST    COMMON    MULTIPLE. 

Art.  107. — A  multiple  of  a  quantity  is  that  which  contains  it 
exactly.  Thus,  6  is  a  multiple  of  2,  or  of  3 ;  and  24  is  a  multi- 
ple of  2,  3,  4,  &c.;  also,  Sd'b^  is  a  multiple  of  2a,  of  2d',  o{2d% 
&c. ;  and  4(a — x)y''  is  a  multiple  of  {a—x),  of  2?/,  of  4?/-,  &c. 

Art.  108. — A  quantity  that  contains  two  or  more  quantities 
exactly,  is  a  common  multiple  of  them.  Thus,  12  is  a  common 
multiple  of  2  and  8  ;  and  dax  is  a  common  multiple  of  2,  3,  a. 
and  X. 


LEAST   COMMON   MULTIPLE.  81 

Art.  109. — The  least  common  multiple  of  two  or  more  quanti- 
ties, is  the  least  quantity  that  will  contain  them  exactly.  Thus, 
6  is  the  least  common  multiple  of  2  and  3  ;  and  \Qxy  is  the  least 
common  multiple  of    2x  and  by. 

Remark. — Two  or  more  quantities  can  have  but  one  least  common 
multiple,  while  they  may  have  an  unlimited  number  of  common  multiples. 
Thus,  while  6  is  the  least  common  multiple  of  2  and  3,  any  multiple  of  6, 
for  instance,  12,  18,  24,  &c.,  will  be  a  common  multiple  of  these  numbers. 

Art.  1 10. — To  find  the  least  common  multiple  of  two  or  more 
quantities. 

It  is  evident,  that  one  quantity  will  not  contain  another  exactly, 
unless  it  contains  the  same  prime  factors.  Thus,  30  does  not  ex- 
actly contain  14,  because  30=2x3X5,  and  14=2X'7  ;  the  prime 
factor  7,  not  being  one  of  the  prime  factors  of  30. 

Art.  111. — Any  quantity  will  contain  another  exactly,  if  it 
contains  all  the  prime  factors  of  that  quantity.  Thus,  30  con- 
tains 6  exactly,  because  30=2X3X5,  and  6=2X3;  the  prime 
factors  2  and  3  of  the  divisor,  being  also  factors  of  the  dividend. 
Hence,  in  order  that  one  quantity  shall  contain  another  exactly,  it 
is  only  necessary  that  it  should  contain  all  the  prime  factors  of 
that  quantity.  Moreover,  in  order  that  any  quantity  shall  exactly 
contain  two  or  more  quantities,  it  must  contain  all  the  different 
prime  factors  of  those  quantities.  And,  to  be  the  least  quantity 
that  shall  exactly  contain  them,  it  should  contain  these  different 
prime  factors  only  once,  and  no  other  factors  besides.  Hence, 
the  least  common  multiple  of  two  or  more  quantities,  contains  all  the 
different  prime  Jaciors  of  these  quaniities  once,  and  does  not  contain 
any  other  factor.  , 

Thus,  the  least  common  multiple  of  d^hc  and  acx,  is  a^bcx,  since 
it  contains  all  the  factors  in  each  of  these  quantities,  and  does  not 
contain  any  other  factor. 

With  this  principle,  let  us  find  the  least  common  multiple  of  ax, 
bx,  and  abc. 

Arranging  the  quantities  as  in  the  margin,  Ave 
sec,  that  «  is  a  fiictor  common  to  two  of  the 
terms  ;  hence  it  must  be  a  factor  of  the  least 
common  multiple,  and  wc  place  it  on  the  left 
of  the  quantities.  We  then  cancel  this  factor 
in  each  of  the  quantities  in  which  it  is  found,  which  is  done  by 
dividing  by  it.  By  examining  the  remaining  factors,  it  is  seen 
that  x  is  a  common  factor  in  the  first  and  second  terms.  We  then 
place  it  on  the  left,  and  cancel  it  in  those  terms  in  which  it  is 


ax 

bx 

abc 

X 

bx 

be 

I 

b 

be 

1 

1 

c 

82  EAY'S   ALGEBRA,    PART   FIRST. 

found.  We  next  see,  that  6  is  a  factor  common  to  two  of  the  quan- 
tities ;  hence,  as  before,  we  place  it  on  the  left,  and  cancel  it  in 
those  terms  in  which  it  is  found.  We  thus  find,  that  a,  x,  b,  and 
c,  are  all  the  prime  factors  in  the  given  quantities  ;  therefore,  their 
product,  abcx,  will  be  the  least  common  multiple  of  these  quanti- 
ties.    Hence,  the 

RULE, 

FOR    FINDING    THE    LEAST   COMMON    MULTIPLE    OF    TWO   OR    MORE 
QUANTITIES. 

1st.  Arrange  ilie  quantities  in  a  liorizontal  line,  and  divide  them 
hij  any  prime  factor  that  will  divide  tioo  or  more  of  them  tciihont  a 
remainder,  and  set  the  quotients,  together  with  the  undivided  quanti- 
ties, in  a  line  beneath. 

2d.  Continue  dividing  as  before,  until  no  prime  factor,  except 
unity,  will  divide  two  or  more  of  the  quantities,  without  a  remainder. 

3d.  Multiply  the  divisors  and  the  quantities  in  the  last  line  together, 
and  the  product  will  be  the  least  common  multiple  required. 

Or,  separate  the  given  quantities  into  their  prime  factors,  and  then 
multiply  together,  such  of  those  factors  as  are  necessary  to  form  a 
product  that  will  contain  all  the  prime  factors  in  each  quantity ;  this 
product  will  be  the  least  common  multiple  required. 

.  Art.  112.— Since  the  greatest  common  divisor  of  two  quan- 
tities, contains  all  the  factors  common  to  them,  it  follows,  that  // 
ive  divide  the  prodiwt  of  two  quantities,  by  their  greatest  common 
divisor,  the  quotient  ivill  be  their  least  common  multiple. 

Find  the  least  common  multiple  in  each  of  the  following  ex- 
amples. 

1.  4a'^,  3a^x,  and  Gaa^'y Ans.  \2a^xhf. 

2.  I2d'x\  Ga\  and  Sx'y- Ans.  24«='.r*?/-. 

3.  Gc'nz',  9n%  and  I2c'nh^ Ans.  SiJc'n'z\ 

4.  15,  Gxz\  9x-V,  and  IScx^ Ans.  QOcx^^*. 

5.  6a Vy,  and  8d\a+x) Ans.  24a*xhj{a-\-x), 

6.  4a'{a—x),  and  Qax^ia^'—x') Ans.  12aVK— x^). 

7.  8x'{x-y),  3aV,  and  I2axy' Ans.  2ia'xY{x-y). 

8.  I0a:'x''{x~y),  lb3^{x-\-y),  and  I2{x'—y-).  A.  ()Od'x^{x'—y''). 

Keview. — 107.  AVhat  is  a  multiple  of  a  quantity?  Give  an  example. 
108.  What  is  a  common  multiple  of  two  or  more  quantities?  Give  an  ex- 
ample. 109.  What  is  the  least  common  multiple  of  two  or  more  quantities  '! 
Give  an  example.  How  many  common  multiples  may  a  quantity  have  ? 
110.  When  is  one  quantity  not  contained  exactly  in  another?  Give  an  ex- 
ample. 111.  When  iii  one  quantity  contained  in  another  exactly  ?  Give 
an  example.  What  is  necessary,  in  order  that  one  quantity  may  exactly 
contain  two  or  more  quantities  ? 


ALGEBRAIC   FRACTIONS.  83 


CHAPTER  III. 

ALGEBRAIC    FRACTIONS. 

DEFINITIONS    AND    FUNDAMENTAL    PROPOSITIONS. 

Art.  113. — If  a  unit,  or  whole  thing,  is  divided  into  any  num- 
ber of  equal  parts,  one  of  the  parts,  or  any  number  of  them,  is 
called  a  fraction. 

Thus,  if  the  line  A  B  hQ  supposed  c       d       e 

to  represent  one  foot,  and  be  divided  ^  |— — !— |— !— — |  B 
into  four  equal  parts,  one  of  those  parts,  as  Ac,  is  called  one 
fourth  (I) ;  tM^o  of  them,  as  Ad,  are  called  two  fourths  (|);  and 
three  of  them,  as  Ae,  are  called  three  fourths  (|). 

In  the  algebraic  fraction  -,  if  c=4  and  1  denotes  1  foot,  then- 
c  c 

denotes  one  fourth  of  a  foot.     In  the  fraction  -,  if  a=3  and  -==-: 

c  c     4 

of  a  foot,  then  -  represents  three  fourths  (f)  of  a  foot. 

Art.  114. — Every  quantity  not  expressed  under  the  form  of  a 
fraction,  is  called  an  entire  algebraic  quantity.  Thus,  «x+6  is  an 
entire  quantity. 

Art.  115.— Every  quantity  composed  partly  of  an  entire  quan- 
tity and  partly  of  a  fraction,  is  called  a  mixed  quantity.     Thus, 

a-\ — ,  is  a  mixed  quantity. 

Art.  116.— An  improper  algebraic  fraction  is  one  whose  nu- 
merator can  be  divided  by  the  denominator,  either  with  or  without 

a  remainder.     Thus,  — ,  and ,  are  improper  fractions. 

a  X 

\  a       c 
Art.  117.— a  single  expression,  as^,  r,  or  -,  is  called  a  sm^Ze 

fraction.     It  may  be  either  proper  or  improper. 

Review. — 111.  What  is  necessary,  in  order  that  any  quantity  may  bo 
the  least,  that  shall  contain  two  or  more  quantities  exactly?  What  fac- 
tors does  the  least  common  multiple  of  two  or  more  quantities  contain  ? 
What  is  the  rule  for  finding  the  least  common  multiple  of  two  or  more 
quantities  ?  IIow  may  the  least  common  multiple  of  two  or  more  quantities 
bo  found,  by  separating  them  into  factors?  112.  If  the  product  of  two 
quantities  be  divided  by  their  greatest  common  divisor,  what  will  the  quo- 
tient be?  113.  What  is  a  fraction?  114.  What  is  an  entire  algebraic 
quantity?  Give  an  example.  115.  What  is  a  mixed  quantity?  Give  an 
example.     116.  What  is  an  improper  algebraic  fraction?     Give  an  example. 


84  RAY'S   ALGEBRA,    PART    FIRST. 

Art.  lis* — A  fraction  of  a  fraction,  as  ^  of  .-r,  or  —  of  7,  is 

2       6         n        G 

called  a  compound  fraction. 

Art.  119. — When  a  fraction  has  a  fraction,  either  in  its  numera- 
tor, or  in  its  denominator,  or  in  both  of  them,  it  is  called  a  comjylex 

fraction.      Thus,  — r'   -r,  — ^ — »  and ,  are  complex  fractions. 

Art.  130. — Algebraic  fractions  are  represented  in  the  same 
manner  as  common  fractions  in  Arithmetic.  The  number  or 
quantity  below  the  line,  is  called  the  denominator,  because  it  de- 
nominates, ov  shows  the  number  of  parts  into  which  the  unit  is 
divided  ;  and  the  number  or  quantity  above  the  line,  is  called  the 
numerator,  because  it  numbers,  or  shows  how  many  parts  are  taken. 

Thus,  in  the  fraction,  -4 ,  the  denominator,  4,  shows,  that  the  unit 
(for  instance,  1  foot,)  is  divided  into  4  equal  parts,  and  the  nume- 
rator, 3,  shows,  that  3    of  these  parts  are  taken.     Again,  in  the 

fraction  -,  the  denominator  c,  shows,  that  a  unit  is  divided  into  c 
c 

equal  parts,  and  a  shows,  that  a  of  these  parts  are  taken. 

The  numerator  and  denominator,  are  called  the  terms  of  a 
fraction. 

Art.  121. — In  the  preceding  definitions  of  numerator  and  de- 
nominator, reference  is  had  to  a  unit  only.  This  is  the  simplest 
method  of  considering  a  fraction;  but  there  is  another  point  of 
view,  in  which  it  is  proper  to  examine  it. 

If  it  be  required  to  divide  3  apples  equally,  between  4  boys,  it 
can  be  effected,  by  dividing  each  of  the  3  apples  into  4  equal 
parts,  and  then  giving  to  each  boy  3  of  those  parts,  expressed  by 
|.  Now,  the  parts  being  equal  to  each  other  in  size,  it  will  be  the 
same,  for  an  individual  to  receive  3  parts  from  1  apple,  or  1  part 
from  each  of  the  3  apples ;  that  is,  |  of  one  apple,  is  the  same  as 
I  of  3  apples;  or,  |  of  1  unit,  is  the  same  as  •]  of  3  units.  Thus, 
§  may  be  regarded  as  expressing  two  fifths  of  one  thing,  or  one 
fifth  of  two  things. 

Review. — 117.  What  is  a  simple  fraction?  Give  an  example.  118. 
What  is  a  compound  fraction  ?  Give  an  example.  119.  What  is  a  complex 
fraction  ?  Give  an  example.  120.  In  Algebraic  Fractions,  what  is  the 
quantity  below  the  line  called?  Why?  Above  the  line?  Why?  Give 
an  example.     What  do  you  understand  by  the  terms  of  a  fraction? 


ALGEBRAIC   FRACTIONS.  85 

So,  —  is  either  the  fraction  -  of  one  unit  taken  m  times,  or  it 
n  n 

is  the  ?ith  of  m  units.     Hence,  the  numerator  may  be  regarded,  as 

showing  the  number  of  units  to  be  divided  ;  and  the  denominator, 

as  showing  the  divisor,  or  what  part  is  taken  from  each. 

Note  to  Teachers  . — Although  it  is  important  that  the  pupil  should 
be  perfectly  familiar  with  the  principles  contained  in  the  foUoAving  proposi- 
tions, the  demonstrations  may  be  omitted,  especially  by  the  younger  class 
of  pupils,  until  the  book  is  reviewed. 

PROPOSITIOi\    I. 

Art.  122* — If  ice  multiply  the  numerator  of  a  fraction,  without 
changing  the  denominator,  the  value  of  the  fraction  is  increased  as 
many  times  as  there  are  units  in  the  multiplier. 

If  we  multiply  the  numerator  of  the  fraction  f  by  3,  without 
changing  the  denominator,  we  get  f.     Thus: 
2X3_6 
7      ~7 

Now,  f  and  f  have  the  same  denominator,  and,  therefore  ex- 
press parts  of  the  same  size;  but  the  second  fraction,  f,  has  three 
times  as  large  a  numerator  as  the  first,  f ;  it  therefore  expresses 
three  times  as  many  of  those  equal  parts  as  the  first,  and  is,  con- 
sequently, three  times  as  large.  And  the  same  may  be  shown  of 
any  fraction  whatever. 

PROPOSITIOiV    II. 

Art.  123* — If  ice  divide  the  numerator  of  a  fraction,  withoid 
changing  the  denominator,  the  value  of  the  fraction  is  diminished, 
as  many  times  as  there  are  units  in  the  divisor. 

If  we  take  the  fraction  |,  and  divide  the  numerator  by  2,  with- 
out changing  the  denominator,  we  get  i.     Thus : 
4-^22 
5       ~5 

Now,  4  and  f  have  the  same  denominator,  and,  therefore,  ex- 
press parts  of  the  same  size ;  but  the  numerator  of  the  second 
fraction,  §,  is  only  one  half  as  large  as  the  numerator  of  the  first, 
I ;  it  therefore  expresses  only  one  half  as  many  of  those  equal 
parts  as  the  first,  and  is,  consequently,  only  one  half  as  large. 
And  the  same  may  be  shown  of  other  fractions. 

Re  viEAV. — 121.  In  what  two  different  points  of  vjew  may  every  fraction 
be  regarded?  Give  examples.  122.  How  is  the  value  of  a  fraction  affected 
by  multiplying  the  numerator  only  ?  How  is  this  proposition  proved  ?  123. 
How  is  the  value  of  a  fraction  affected  by  dividing  the  numerator  only  ? 
How  is  this  proposition  proved  ? 


86  EAY'S   ALGEBRA,    PART   FIRST. 


PROPOSITION    III. 

Art.  134* — If  we  multiply  the  denominator  of  a  fraction,  with- 
out changing  the  numerator,  the  value  of  the  fraction  is  diminished^ 
as  many  times  as  there  are  units  in  the  multiplier. 

If  we  take  the  fraction  |,  and  multiply  the  denominator  by  2, 
without  changing  the  numerator,  we  get  f.     Thus ; 
3  3 

4X2~8 

Now,  each  of  the  fractions,  |  and  §,  have  the  same  numerator, 
and,  therefore,  express  the  same  number  of  parts ;  but,  in  the 
second,  the  parts  are  only  one  half  the  size  of  those  in  the  first ; 
consequently,  the  whole  value  of  the  second  fraction,  is  only  one 
half  that  of  the  first.  And  the  same  may  be  shown  of  any  frac- 
tion whatever. 

PROPOSITION     IV. 

Art.  125. — If  we  divide  the  denominator  of  a  fraction,  without 
changing  the  numerator,  the  value  of  the  fraction  is  increased  as 
many  times  as  there  are  units  in  the  divisor. 

If  we  take  the  fraction  |,  and  divide  the  denominator  by  3, 
without  changing  the  numerator,  we  get  f .     Thus: 
2  2 

9h-3~3 
Now,  each  of  the  fractions,  |  and  |,  have  the  same  numerator, 
and,  therefore,  express  the  same  number  of  parts;  but,  in  the 
second,  the  parts  are  three  times  the  size  of  those  of  the  first ; 
consequently,  the  whole  value  of  the  second  fraction  is  three  times 
that  of  the  first.     And  the  same  may  be  shown  of  other  fractions. 

PROPOSITION    V. 

Art.  136. — Multiplying  both  terms  of  a  fraction  hy  the  same 
number  or  quantity,  changes  the  form  of  the  fraction,  bid  does  not 
alter  its  value. 

If  we  multiply  the  numerator  of  a  fraction  by  any  number,  its 
value  (by  Prop.  I.)  is  increased,  as  many  times  as  there  are  units 
in  the  multiplier;  and,  if  we  multiply  the  denominator,  the  value 
(by  Prop.  III.)  is  decreased,  as  many  times  as  there  are  units  in 
the  multiplier.  Hence,  if  both  terms  of  a  fraction  are  multiplied 
by  the  same  number,  the  increase  from  multiplying  the  numerator, 

Review. — 124.  How  is  the  value  of  a  fraction  aifected  by  multiplying 
only  the  denominator  ?  How  is  this  proposition  proved  ?  125.  How  is  the 
value  of  a  fraction  affected  by  dividing  the  denominator  only  ?  How  is  this 
proposition  proved  ?  126.  How  is  the  value  of  a  fraction  affected  by  mul- 
tiplying both  terms  by  the  same  quantity  ?     Why? 


ALGEBRAIC    FRACTIONS.  87 

is  equal  to  the  decrease  from  multiplying  the  denominator ;  con- 
sequently, the  value  remains  unchanged. 

PROPOSITIOIV  VI. 

Art.  127»— Dividing  both  terms  of  a  fraction  by  the  same  num- 
ber or  quantity,  changes  the  form  of  the  fraction,  but  does  not  alter 
its  value. 

If  we  divide  the  numerator  of  a  fraction  by  any  number,  its 
value  (by  Prop.  II.)  is  decreased,  as  many  times  as  there  are  units 
in  the  divisor ;  and  if  vs-e  divide  the  denominator,  the  value  (by 
Prop.  IV.)  is  increased,  as  many  times  as  there  are  units  in  the 
divisor.  Hence,  if  both  terms  of  a  fraction  are  divided  by  the 
same  number,  the  decrease  from  dividing  the  numerator  is  equal 
to  the  increase  from  dividing  the  denominator ;  consequently,  the 
value  remains  unchanged. 

CASE  Iw 

TO    REDUCE    A    FRACTION    TO    ITS    LOWEST    TERMS. 

Art.  12S. — Since  the  value  of  a  fraction  is  not  changed  by 
dividing  both  terms  by  the  same  quantity  (See  Art.  127),  Ave  have 
the  following 

RULE. 

Divide  both  terms  by  their  greatest  common  divisor. 
Or,  Resolve  the  numerator  and  denominator  into  their  prime  fac- 
tors, and  then  cancel  those  factors  common  to  both  terms. 

Remark  . — The  last  rule  -will  be  found  most  convenient,  when  one  or 
both  terms  are  monomials. 

AaW 
1.  Reduce  ^7-^  to  its  lowest  terms. 

4ab^  _2abX2b_2ab    . 
6bx'~~Sx'X2b~Sx' 
Reduce  the  following  fractions  to  their  lowest  terms. 


^-  -W 3«- 

„    6aV  .       3a 

3-8^ ^"^-ii- 

8a%*  4?/* 

5.    ,..  .r.T-  .    .    .  Ans.    - 


8a''b  .  2a 

7.  T^r-i  9  ,  A   1    •    Ans. 


\2ab''+4abc'  '  Sb-^c' 

^    2a^cx^-\-2acx  .        ax-\-l 

8.  — i-7T-4 •  Ans.  — =-- . 

1  Oac^x  oc 

6a''b+ba¥  ^^^  a+b 

babc-\-babd'  '          '  c-\-d 


VZx^y^z^'  '    '  4y 

Eeview. — 127.  How  is  the  value  of  a  fraction  affected  by  dividing  both 
terms  by  the  same  quantity  ?  Why  ?  128.  How  do  you  reduce  a  fraction 
to  its  lowest  terms  ? 


10. 

11. 

12. 


RAY'S    ALGEBRA,    PART   FIRST. 
56xh/ 


24xy 


2-— 40x3/^ 
6ac 


I2a'c'—I8ac'' 
12x\i/—lSxf 


Ans.  7 
Ans. 
Ans 


7x_ 
Sx — 5?/* 
1 

2ac — 3c* 
2x-Si/ 


I8x'!/+I2xif ^""' Sx-\-2!/' 

Note.  — In  the  preceding  examples,  the  greatest  common  divisor  in  each 
is  a  monomial ;  in  those  which  follow,  it  is  a  polynomial ;  but,  by  separating 
the  quantities  into  factors,  or  by  the  rule  (Art.  106,)  the  greatest  common 
divisor  is  readily  found. 

lo.    r-   ,  ;  f-,..   .     Ihis  IS  equal  to 


bab+^b-' 


3a{c 


14. 
15. 
16. 
17. 

18. 


56(a+6) 
Sz'-24z-{-   9 


4z'~S2z+l2'     ^'''^•4 


3a{a-{-b){a—b)__Sa{a—b) 
bb{a+b)      ~~      56~" 


5a^+5ax 

a'—x'    ' 

n'—2n+l 

I4d'—7ab 
lOac — 56c' 
x^—xif 
x*—y* 


Ans. 
Ans. 
Ans 
Ans 


5a 


a — X 

Itr-l 

7a 
5c' 

X 

x'^+f 


19. 
20. 
21. 
22. 
23. 


-6*' 

X'^ — 7/'^ 


x^—2xy+i/ 

x^—ax^ 
x^ — 2ax-j-a^' 
2x'—6x 

7? — X — 6  * 

a;=^+2a;— 15 

x'^+8x  +  15* 


Ans 
Ans. 
Ans. 
Ans. 
Ans. 


1 

x—y 

x' 
X — a 
2x 

x+2' 
x-3 
x-\-3 


Art.  129. — Exercises  in  Division  (See  Art.  76,)  in  which  th( 
quotient  is  a  fraction,  and  capable  of  being  reduced  to  lower  terms 

5a; 

1.  Divide  bx^y  by  3xy^.   » Ans.  ^ 

2.  Divide  15a-^6'''c  by  25a'6c Ans.  ^ 

56 

3.  Divide  25a6c  by  5ac''^ Ans.  — , 


4.  Divide  amn^  by  a^mhi Ans. 


In  a  similar  manner,  when  one  polynomial  can  not  be  exactly  divided  by 
another,  the  division  may  be  indicated,  and  the  result  reduced  to  its  motit 
simple  form. 

5a; 

5.  Divide  25ax^  by  5ax^ — 5axy Ans.  . 

x—y 

6.  Divide  Sm'-hSn'  by  I5m'+I5u' Ans.  \. 

D 


7.  Divide  x^ii'^-\-xh/  by  dx^y-^-axy"^ Ans. 


xy 


REDUCTION    OF   FRACTIONS.  89 

8.  Divide  4a+46  by  2a'^— 26^ Ans.  -^. 

a—b 


9.  Divide  n^ — 2?i^  by  w-'— 4n+4 Ans. ^. 

10.  Divide  a;2-f  2x— 3  by  x^+Sx+G Ans.  ^. 

CASE   II. 

TO    REDUCE    A    FRACTION    TO    AN    ENTIRE    OR    MIXED   QUANTITY. 

Art.  130. — Since  the  numerator  of  the  fraction  may  be  re- 
garded as  a  dividend,  and  the  denominator  as  a  divisor,  this  is 
merely  a  case  of  division.     Hence,  the 

RULE. 

Divide  the  numerator  hy  the  denominator,  for  the  entire  part,  and, 
if  there  he  a  remainder,  place  it  over  the  denominator  for  the  frac- 
tional part. 

Note. — The  fractional  part  should  be  reduced  to  its  loAvest  terms. 

_     ^,    -         Zax+y^  ^  .      ,  ... 

1.  Reduce to  a  mixed  quantity. 

X 

3«X+62  7,2 

—  =3a+- .    Ans. 

X  X 

Reduce  the  following  fractions  to  entire  or  mixed  quantities. 

2    ^^J        Ans.6+^-. 

a  .  « 

3.  £^1' Ans.  c-d. 

d 

4.2!±^ Ans.a+^+|^. 


5. 


?^!^^-.    .    _. Ans.2ax-?. 


a 


a 


o^-xH-S ^^g  a-x+-4-- 

a+x  a+x 

4ax-2x^-a^ ^^^  2.-^-. 

8  it?^^         Ans.a-^. 

a—x    '  a— X 

9  t±c^_x* A„g,  a^-ax+x'-  -^. 

a+x  «+« 

10    _J2^^_-I^ Ans.  3+^,. 

^"-  4x»— x'^— 4x+l  « -1 

8 


90 


CAS  E    III. 

TO    REDUCE    A    MIXED    QUANTITY    TO    THE    FORM    OF    A    FRACTION. 

Art.  131. — 1.     In  2^  how  many  thirds? 

In  1  unit  there  are  3  thirds ;  hence,  in  2  units,  there  are  twice 
as  many,  that  is,  6 ;  then,  6  thirds  plus  1  third,  are  equal  to  7 

thirds ;  that  is,  2^  are  equal  to  ^.     In  the  same  manner,  a-\ —  is 

w   «^  I  ^      u-  u  •            1  ^    ac+b 
equal  to \--,  which  is  equal  to . 

Hence,  the 

RULE, 

FOR    REDUCING    A    MIXED    QUANTITY    TO    THE    FORM    OF    A    FRACTION. 

Multiply  the  entire  part  by  the  denominator  of  the  fraction  ;  then 
add  the  numerator  with  its  proper  sign  to  the  product,  and  place  the 
residt  over  the  denominator. 

Remark  . — Cases  II.  and  III.,  are  the  reverse  of,  and  mutually  prove 
each  other. 

Before  proceeding  further,  it  is  important  for  the  learner  to 
consider 

THE    SIGNS    OF    FRACTIONS. 

Art.  132.-  It  has  been  already  stated  (See  Art.  121,)  that  in 
every  fraction  the  numerator  is  a  dividend,  the  denominator  a 
divisor,  and  the  value  of  the  fraction  the  quotient.  The  signs  pre- 
fixed to  the  terms  of  a  fraction,  affect  only  those  terms ;  and  the 
sign  placed  before  a  fraction,  aflfects  its  whole  value.     Thus,  in  the 

fraction \ — ,  the  sign  of  a^,  the  first  term  of  the  numerator, 

x^y 

is  plus;  of  the  second,  6'^,  minus;  while  the  sign  of  each  term  of 

the  denominator,  is  plus.     But  the  sign  of  the  fraction,  taken  as 

a  whole,  is  minus. 

By  the  rule  for  the  signs  in  Division,  Art.  75,  we  have 

=+&;  or,  changing  the  signs  of  both  terms, =+&• 

~r~tt  — a 

But,  if  we  change  the  sign  of  the  numerator,  we  have  — — = — b. 

-\-a 

And,  if  we  change  the  sign  of  the  denominator,  we  have  = — b. 

Hence,  the  signs  of  both  terms  of  a  fraction  may  be  changed, 
without  altering  its  value,  or  changing  its  sign  ;  but,  if  the  sign  of 
either  term  of  a  fraction  be  changed,  and  not  that  of  the  other,  the 
sign  of  the  fraction  will  be  changed. 


REDUCTION   OF   FRACTIONS.  91 

From  this,  it  also  follows,  that  the  signs  of  either  term  of  a  frac- 
tion may  he  changed,  without  altering  its  vahie,  if  the  sign  of  tlie 
fraction  he  changed  at  the  same  time. 

_,,  ax — x^        ax—x^        x^ — ax 

Thus,  .... = = . 

c  —  c  c 

.     ,  a — x        ,  a — X        ,  X — a 

And,    .    ,    .     a —-=a-{ 7=aH — -. — . 

0  — 0  0 

EXAMPLES. 

1.  Reduce  3a-\ to  a  fractional  form. 

X 

o       Sax       ,  Sax  ,  ax — a    Sax4-ax — a    4ax — a 

Sa= and = = .     Ans. 

XXX  X  X 

2.  Reduce  4a ^ —  to  a  fractional  form. 

Sc 

.       I2ac       -  I2ac    a—b     \2ac — [a — h)     \2ac—a-\-h       . 

4a=-^—  and  -^ ^r— = ^-^^ ^= ^ .     Ans. 

6c  Sc         Sc  Sc  Sc 

Remark. — In  solving  this  example,  tho  learner  should  observe,  that 
a — h 

— ;t —  is  to  be  subtracted  from  4«.  We  reduce  Aa  to  a  quantity  whose  de- 
nominator is  3c ;  then  make  the  subtraction,  and  write  the  result  over  the 
common  denominator,  3c. 

Reduce  the  following  quantities  to  improper  fractions. 

„    -    ,  a—h  .        \Ocx-\-a — h 

^-  ^'+^ ^"^- 2£— 

.     -       a — h  .        ]Ocx—a-\-b 

4.  5c T^— Ans. ^ . 

2x  2x 

c—d  J.        Sxhj-\-c—d 

5.  Sx-\ Ans. ~ . 

xy  X]/ 

,.    „      4x'-5  .        Ux'+5 

6.  3a; = Ans.  — = . 

5x  ox 

rs,+^ Ans.^:±L« 

•         5?/  5?/ 

„        ,       ,      X                                                             .^^   3c'+2xi/-\-if+x 
8.  x+yH — : — Ans.  ■ . 

«-i+i^: ^-&i- 

Review. — 130.  How  do  you  reduce  a  fraction  to  an  entire  or  mixed 
quantity  ?  131.  How  do  you  reduce  a  mixed  quantity  to  the  form  of  a 
fraction  ?  132.  What  do  the  signs  prefixed  to  tho  terms  of  a  fraction 
affect?  What  does  the  sign  placed  before  the  whole  fraction,  affect  ?  What 
effect  does  it  have  upon  the  value  of  a  fraction,  or  upon  its  sign,  to  change 
the  signs  of  both  terms  ?  To  change  the  sign  or  signs  of  one  term,  and  not 
of  the  other  ?     To  change  the  sign  of  the  fraction,  and  one  of  its  terms  ? 


92  RAY'S   ALGEBRA,    PART   FIRST.  ^ 

10.  -.-^. 5 Ans. . 

2x-\-z  Zx-\-z 

11.  ?±-^i:-6 Ans.5£=i5 

12.  3a'.- «-^^"- Am.?^'±^. 

X  X 

U.  a-\-x-\ Ans, . 

a—x  a—x 

14.  xy'-5fc^ An«.^. 

X  ■  X 

15.  a^~x' ::rT-~i -^"s. TT~->- 

...             ,  a?-^x'-b                                                    .         2«'^-5 
lb.  a — xA ; Ans.  — . 

a-\-x  a-\-x 

17.  a^ — d^x-\-ax^ — x^ — — |-^ Ans. . 

a+x  a-\-x 

CASE   IV. 

TO    REDUCE    FRACTIONS    OF  DIFFERENT    DENOMINATORS    TO    EQUIVALENT 
FRACTIONS,    HAVING    A    COMMON    DENOMINATOR. 

Art.  133.— 1.  Reduce  -r  and  -  to  a  common  denominator. 
0  a 

If  we  multiply  both  terms  of  the  first  fraction,  -,  by  d,  the  de- 
nominator of  the  second,  we  shall  have  -=-——=;  — - ;  and,  if  we 

h    bXd     bd 
c 
multiply  both  terms  of  the  second  fraction,  -,  by  6,  the  denomina- 
tor of  the  first,  we  shall  have  -=--— y=--, 

d    dXb    bd 

In  this  solution  we  observe ;  ^firsi,  the  values  of  the  fractions 
are  not  changed,  since,  in  each  fraction,  both  terms  are  multiplied 
by  the  same  quantity ;  and,  second,  the  denominators  in  each 
must  be  the  same,  since  they  consist  of  the  product  of  the  same 
quantities. 

2.  Reduce  — ,  -,  and  -,  to  a  common  denominator. 
in    n  r 

Ilore,  we  are  at  liberty  to  multiply  both  terms  of  each  fraction, 
by  the  same  quantity,  since  this  (See  Art.  120)  will  not  change 
its  value.  Now,  if  we  multiply  both  terms  of  each  fraction,  by 
the  denominators  of  the  other  two  fractions,  the  new  denomina- 
tors in  each  will  be  the  same,  since,  in  each  case,  they  will  consist 
of  the  product  of  the  same  factors,  that  is,  of  all  the  denominators. 

I 


REDUCTION   OF   FRACTIONS.  93 

Thus, ._aXnXr_  anr 

»*XwX^    'mnr 

by^myCr bmr 

ny<,'mXr~mni'' 

ryjaXn    mnr 

It  is  evident,  that  the  value  of  each  fraction  is  not  changed,  and 
that  they  have  the  same  denominators.     Hence,  the 

RULE, 

FOR    REDUCING    FRACTIONS    TO    A    COMMON    DENOMINATOR. 

Multiply  both  terms  of  eacli  fraction  by  the  jyroduct  of  all  the 
denominators,  except  its  own. 

Remark. — Since  each  denominator  of  the  new  fractions,  will  consist 
of  the  product  of  all  the  denominators  of  the  given  fractions,  it  is  unnec- 
essary to  perform  the  same  multiplication  more  than  once. 

EXAMPLES. 

Reduce  the  following  fractions,  in  each  example,  to  others, 
having  a  common  denominator. 

_    a  c       ^  1  .       2ad  2bc        ,  bd 

^-  &'rr"^^2 ^''''2bd^2bd^^''^2b7r 

4.^,and^ Ans.^,and^^^±^. 

y  c  cy  cy 

^    2  3a       ,  x~y  ,        86    9a&        ,  12x-12y 

5.  3,-^,  and --^ ^"^•I26'r26'""^— 126— • 

^    2a:  3a;        -  .         lOxz    9xy         ,  \bayz 

3y'  52  15?/z'  15^2           15yz 

^    a  X        ,  ri  .         ayz    xh        ,  xtf 

7.  -,  -,  and  - Ans.  — -, ,  and  — ^-. 

X  if         z  xyz    xyz  xyz 

^    I    x'        ,  x'-hz'  ,        3a:+3z  2r'+2xh        ,  6x'+6z' 

8.  ci,  -o->  and ; — .     .      Ans.  ^ — -jr,  -n — rn — >  and  -; — -77-. 

2    3  x+z  6a;+62    bx-r^z  Ux+Kiz 

^    x+y       ,  x—y  .        x^+2xy-^tf       ,  x'^—2xy+7f 

9.  —^,  and  —-^ Ans. t-^V^'  ^"^ 2 J—- 

x—y  x+y  x'-y'  x'—y' 

36  .       ac  36  cd       ,  5c 

10.  a,  —,  d,  andb Ans.  — ,  — ,  — ,  and  — . 

Re  VIE  w. — 133.  How  do  you  reduce  fractions  of  different  denominators 
to  equivalent  fractions  having  the  same  denominator  ?  Why  is  the  value 
of  each  fraction  not  changed  by  this  process?  Why  does  this  process  give 
to  each  fraction  the  same  denominator? 


94  RAY'S  ALGEBRA,   PART   FIRST. 


,  -      a      m — n       ,      a 
11.  -i^-, ,  and ■ — . 

oni       a  m-\-n 


a'^m-\-ahi         3wi^ — Smn^         ,         3a 


7)1 


Ans.  5 r— vj ,  ^ .-— 7j ,  and  o      •,  i  o 

oanr-f-Samn  Sanr-jroanm  Sanr-\-oamn 

Art.  134* — It  frequently  happens,  that  the  denominators  of 
the  fractions  to  be  reduced,  contain  one  or  more  common  factors. 
In  such  cases,  the  preceding  rule  does  not  give  the  least  common 
denominator.  From  the  preceding  Article  we  see,  that  the  com- 
mon denominator  is  a  multiple  of  all  the  denominators  ;  and,  that 
each  numerator  is  multiplied  by  a  quantity  which  is  equal  to  the 
quotient  obtained,  by  dividing  this  multiple  by  its  denominator. 
Thus,  in  the  second  example,  nr,  mr,  and  mn,  the  quantities  by 
which  each  numerator  is  respectively  multiplied,  may  be  regarded 
as  the  quotients  obtained,  by  dividing  mnr  successively,  by  m,  n, 
and  r.  Noav,  if  we  obtain  the  least  common  multiple  of  the  de- 
nominators, 1)y  the  rule,  Case  III.,  and  then  divide  it  by  each 
denominator  respectively,  and  multiply  the  quotients  by  the  nu- 
merators respectively,  we  shall  obtain  a  new  class  of  fractions, 
equivalent  to  the  former,  and  having  for  a  common  denominator, 
the  least  common  multiple  of  the  given  denominators.  It  is  easily 
Reen.  that  both  terms  of  each  fraction  are  multiplied  by  the  same 
quantity,  and  hence,  that  the  resulting  fractions  are  equivalent  to 
the  given  ones. 

1.  Reduce  -j,  y-,  and  — ,  to  equivalent  fractions,   having  the 

least  common  denominator. 

The  least  common  multiple  of  the  denominators  is  easily  found 
to  be  hcd;  dividing  this  by  6,  the  denominator  of  the  first  fraction, 

the  quotient  is  cd ;  then  multiplying  both  terms  of  -j  by  cd,  the 

.,  .  mcd 

result  IS -z — r . 

ocd 

Then  bcd-r-bc=d,  and  ^— ,   -= -=—:,. 

ocY.d  bed 

Also,  bcd-~-cd=b,  and  — ,,   -= - — r. 

cdXb  bed 

The  process  of  multiplying  the  denominators  by  the  quotients 
may  be  omitted,  as  the  product  in  each  case  will  be  equal  to  the 
least  common  multiple.     Hence,  the 

RULE, 

FOR   REDUCING    FRACTIONS    OF    DIFFERENT    DENOMINATORS,    TO   EQUIVA- 
LENT   FRACTIOxNS,    HAVING    THE    LEAST    COMMON    DENOMINATOR. 

1st.  Find  the  least  common  midtiple  of  all  the  denominators; 
this  will  be  the  common  denowinaior. 


REDUCTION   OF   FRACTIONS.  95 

2d.  Divide  the  least  common  multiple,  hy  the  first  of  the  given 
denominators,  and  multiplij  the  quotient  hij  the  first  of  the  given 
numerators;  the  product  loill  he  the  first  of  the  required  numerators. 

3d.  Proceed,  in  a  similar  manner,  to  find  each  of  the  other 
numerators. 

Note. — Each  fraction  should  bo  in  its  lowest  terms, before  commencing 
the  operation. 

Reduce  the  following  fractions,  in  each  example,  to  equivalent 
fractions,  having  the  least  common  denominator. 

^     2a_   3x           _%  Aad    IShx  bey 

36?  Vd'         6bd ^°^-  66^'  66^r  ^""^  66^' 

„     m      n         J     r  b^cdm      acdn  .    ahh' 

4.   =^+1,  ^,  and  ?;+<.  .    .  An..  M,  fc*'-,  and  ^^4 
X — y  x-f-y  X- — y^  x^ — y^     x'- — y'-  x^ — y^ 

Other  exercises  will  be  found  in  the  addition  of  fractions. 

Note. — The  two  following  Articles  depend  on  the  same  principle  as  the 
two  preceding,  and  are,  therefore,  introduced  here.  They  will  both  bo 
found  of  frequent  use,  particularly  in  completing  the  square,  in  the  solution 
of  equations  of  the  second  degree. 

Art.  135. — To  reduce  an  entire  quantity  to  the  form  of  a  frac- 
tion having  a  given  denominator. 

1.  Let  it  be  required  to  reduce  a  to  a  fraction  having  b  for  its 
denominator. 

Since  any  quantity  may  be  reduced  to  the  form  of  a  fraction, 

by  writing  1  beneath  it,  a  is  the  same  as  y  ;  if  we  multiply  both 
terms  by  b,  which  will  not  change  its  value  (See  Art.  126),  wc 
have  1  ■=y->  for  the  required  fraction.     Hence,  the 

RULE, 

FOR    REDUCING    AN    ENTIRE    QUANTITY    TO    THE    FORM    OF    A    FRACTION 
HAVING    A   GIVEN    DENOMINATOR. 

Multiply  the  entire  quantity  by  the  given  denominator,  and  write 
the  product  over  it. 

EXAMPIiES. 

2.  Reduce  x  to  a  fraction,  whose  denominator  is  4.        Ans.   -r. 

3.  Reduce  m  to  a  fraction,  whose  denominator  is  9a^. 

Ans.  ?^\ 

Review. — 134.  How  do  you  reduce  fractions  of  different  denominators 
to  equivalent  fractions,  having  the  lea»t  common  denominator? 


96  RAY'S   ALGEBRA,    PART   FIRST. 


4.  Reduce  3c+5  to  a  fraction  whose  denominator  is  16cl 

Ans.  — ,-^-r, —  . 

5.  Reduce  a—b  to  a  fraction,  whose  denominator  is  a^ — 2a6+6l 

a:'—2ab-i-b-'  {a— by' 

Art.  136. — To  convert  a  fraction  to  an  equivalent  one,  having 
a  denominator  equal  to  some  multiple  of  the  denominator  of  the 
given  fraction. 

1.  Reduce  y  to  a  fraction,  whose  denominator  is  be. 

b 

It  is  evident,  that  the  terms  must  be  multiplied  by  the  same 
quantity,  so  as  not  to  change  the  value  of  the  fraction.  It  is  then 
required  to  find,  what  the  denominator,  b,  must  be  multiplied  by, 
that  the  product  shall  become  be ;  but,  it  is  evident,  this  multi- 
ple will  be  found,  by  dividing  be  by  b,  which  gives  the  quotient,  c. 

Then,  multiplying  both  terms  of  the  fraction  -  by  c,  the  result  is 

J—,  which  is  equal  to  the  given  fraction  -,  and  has,  for  its  denom- 
inator be.     Hence,  the 

RULE, 

FOR    CONVERTING    A    FRACTION     TO     AN    EQUIVALENT     ONE,    HAVING    A 
GIVEN    DENOMINATOR. 

Divide  tlie  given  denominator  by  the  denominator  of  the  given 
fraction,  and  inultiply  both  terms  by  the  quotient. 

Pt  E  M  A  u  K. — This  rule  is  perfectly  general,  but  it  is  never  applied,  except 
where  the  required  denominator  is  a  multiple  of  the  given  one.  In  other 
cases,  it  would  produce  a  complex  fraction.  Thus,  if  it  is  required  to  con- 
vert i  into  an  equivalent  fraction,  whose  denominator  is  5,  the  numerator 
of  the  new  fraction  would  bo  2^. 

2.  Convert  j  to  an  equivalent  fraction,  having  the  denomina- 
tor 16.  Ans.  YT- 

lb 

3.  Convert  ^  to  an  equivalent  fraction,  having  the  denomina- 
tor  9.  Ans.  -^  . 

4.  Convert  -  to  an  equivalent  fraction,  having  the  denomina- 
tor  aV.  Ans.  -.y-r, . 

Review. — 134.  If  each  fraction  is  not  in  its  lowest  terms,  before  com- 
mencing the  operation,  what  is  to  be  done?  135.  How  do  you  reduce  an 
entire  quantity  to  the  form  of  a  fraction  having  a  given  denominator? 


ADDITION    AND    SUBTRACTION   OF   FRACTIONS.     97 

5.  Convert to  an  equivalent  fraction,  having  the  denomi- 

nator  7n^ — 2mn-\-n^.  Ans. 


6,  Convert  ,        to  an  equivalent  fraction,  having  the  denomi- 

nator  a\h-\-cY.  Ans.     .,      '    :„. 

CASE  V.  a\b-]rcy 

ADDITION    AND    SUBTRACTION    OF    FRACTIONS. 

Art.  137. — 1.  Let  it  be  required  to  find  the  sum  of  §  and^. 
Here,  both  parts  being  of  the  same  kind,  that  is,  fifths,  we  may 
add  them  together,  and  the  sum  is  6  fifths,  (f ). 

2.  Let  it  be  required  to  find  the  sum  of —  and  — . 

mm 

Here,  the  parts  being  of  the  same  kind,  that  is,  wths,  we  may, 
as  in  the  first  case,  add  the  numerators,  and  write  the  result  over 
the  common  denominator. 

Thus, ^+1=^. 

m      m       m 

3.  Again,  let  it  be  required  to  find  the  sum  of  —  and  -. 

^  ^  m  n 

Here,  the  parts  not  being  of  the  same  kind,  that  is,  the  denom- 
inators being  difierent,  we  can  not  add  the  numerators  together, 
and  call  them  by  the  same  name.  We  may,  however,  reduce  them 
to  a  common  denominator,  and  then  add  them  together. 

_,,         a      an      c     cm       .     ,    an  .    cm    an-\-cm 

Thus, — = ;    -= — .     And \ = . 

m     mn      n    mn  mn     mn       mn 

Hence,  the 

RULE, 

FOR    THE    ADDITION    OF    FRACTIONS. 

Reduce  the  fractions y  if  necessary,  to  a  common  denominator; 
add  the  numerators  together,  and  place  their  sum  over  the  common 
denominator. 

Art.  138. — It  is  obvious,  that  the  same  principles  would  apply, 
if  it  were  required  to  find  the  difierence  between  tAVO  fractions ; 
that  is,  if  their  denominators  were  the  same,  the  numerators  might 
be  subtracted ;  but,  if  their  denominators  were  difi'erent,  it  would 
be  necessary  to  reduce  them  to  the  same  denominator,  before  per- 
forming the  subtraction.     Hence,  the 

RULE, 

FOR   THE    SUBTRACTION    OF    FRACTIONS. 

Reduce  the  fractions,  if  necessary,  to  a  common  denominator ; 
then  subtract  the  numerator  of  the  fraction  to  be  subtracted  from  the 
numerator  of  the  other,  and  place  the  remainder  over  the  common 
denominator. 
9 


RAY'S   ALGEBRA,    PART   FIRST. 


EXAMPLES    IN    ADDITION    OP  FRACTIONS. 

4.  Add  ^,  ^,  and  ^,  together Ans.  a. 

5.  Add  U,  -;=,  and  ji  together Ans.  ^-j^. 

6  o  b     ^  10 

6.  Add  -,  T,  and  -  together Ans. -. . 

ah  c      ^  abc 

7.  Add  ^,  ^,  and  2  together Ans. ^2    ""^' 

o     .  ,,   3x  4x        ,  5a;  ^       .  .        143a;    ,.,     ,23a; 

8.  Add  -7-,  -^,  and -TT  together.    .    .    Ans.  -777^=^^+-^/^  • 

4     5  b  oU  bU 

9.  Add      ^     and  — ^  together Ans.  a;. 

•  10.  Add   — -T  and j-  together Ans.  —. — ,t,- 

11.  Add   — ^  and  —^together Ans.  %~-,- 

X-\-7/  X—tJ       *=  x'—lf 

TO     KAA   5+aJ  3— ax        -1   ^    ,       n  A        15a+&?/+9 

12.  Add   , ,  and  rr  together.  .    .  Ans. '    '        . 

y        ay  da     "  day 

13.  Add   — —,  —. — ,  and together Ans.  0. 

ah        be  ac 

14.  Add  Tj— — ,  .j ,  and  =— —  together.     ....    Ans.  .j . 

1+a;   1— X  l+x     °  1 — X 

When  entire  quantities  and  fractions  are  to  be  added  together, 
they  may  be  connected  by  the  sign  of  addition,  or  the  entire  quan- 
tities and  the  fractions  may  be  reduced  to  a  common  denominator, 
and  the  addition  then  performed. 

15.  Add  2a;,  3a;-f--^,  and  «+ q-  together.    .    .    Ans.  63;+-^--. 

16.  Add   5a;H — ^—  and  4a; ^ —  together. 

.        o    ,  5a;*^— 16x+9 
Ans.9x+ ^^-^—. 

if^     1  1 1   o  ,  2a   _     3a — 2a;        .  ^  ,  x — a  , 

17.  Add  3H — ,  5 ,  and  7-| together. 

Ans.  15-^-^^-^'. 
ax 

18.  Add  7,  — —r,  and  2  together Ans.  —, — ,-,. 

a—h  a-\-h  ^  a^—¥ 

Review. — 136.  How  do  you  convert  a  fraction  to  an  equivalent  one, 
having  a  given  denominator  ?  Explain  the  operation  by  an  example.  137. 
When  fractions  have  the  same  denominator,  how  do  you  add  them  together? 
When  fractions  have  different  denominators,  how  do  you  add  them  together  ? 


SUBTRACTION   OF   FRACTIONS.  99 


EXAMPLES    IIV    SUBTRACTION    OF    FRACTIONS. 

1.  From  ^  take  ^ Ans.  ^. 

2.  From  -^  take  ^ Ans.  y^- 

3.  From  — ^  take  — ^ Ans.  b. 

.    _,        ^ax  ^  .     bax  .  Wax 

4.  From -^- take -^- Ans. tv- • 

5.  From -J- take  ,Y- Ans. — -: . 

4a  2x  4ax 

6.  From  -r-  take  ^5- Ans.  — ,7, . 

4x  Sa  r2ax 

7.  From^take^^ Ans.  ^. 

„    ^        a^+ax  ^  ,     a'^—ax  .       2ax'^-\-2a^i/ 

8.  From take  — ; — Ans. 7, ^- . 

x—y  x-\-y  x'—y' 

-    _,        2a+6 ,  ,     2a-h  .        126-a 

9.  From  — = —  take  -^= — Ans.  —^^^ — . 

be  Ic  ODC 

10.  From  5a;+:r  take  2a: .     .    .    .     Ans.  dxH r • 

he  he 

11.  From r  take — -^ Ans. -^j — ^,. 

a—h  a-\-h  a^—¥ 

,,11  ,       a''h^ah''—a—h 

12.  From  a-\-h  take  — Ht Ans. ^ . 

ah  ah 

a:3  1  ^/3  a;3_y3  2j^i/+2xy^ 

13.  From  ^-^^- take  ^^- Ans. — 1^-. 


14.  From  ^^.,  take  ^^:p^~, A°«- IZT^-i- 

15.  From     ^  ,     , -,—  take  -—-r Ans.  -— r. 

a^—¥  a-\-h  a—h 

16.  From  ^.py  take  ^-j ^^TR- 

1  2  ,       a;3— 2x+3 

17.  From  x-{ =-  take  — -r Ans.  — -^— j — . 

a:— 1  x+1  x'—l 

18.  From  2a-3a:+^=^  take  a— 5a;+^=^.    A.  a+2a;H-— -'. 

a  x  ox 

19.  From  a+aJ+^^i  take  a-a:+— .      .     Ans.  2x+^^v 

Review. — 138.  If  two  fractions  have  the  same  denominator,  how  do 
you  find  their  difiFerence  ?  When  two  fractions  have  different  denominators, 
how  do  you  find  their  difiFerence  ? 


100  RAY'S   ALGEBRA,    PART   FIRST. 

CASE  Vli 

TO    MULTIPLY    ONE    FRACTIONAL   QUANTITY    BY    ANOTHER. 

Art.  139. — To  multiply  a  fraction  by  an  entire  quantity,  or 
an  entire  quantity  by  a  fraction. 

It  is  evident,  from  Prop.  I.,  Art.  122,  that  in  multiplying  the 

numerator  of  a  fraction  by   an  entire  quantity,  the  fraction  is 

increased  as  many  times  as  there  are  units  in  the  multiplier. 

mi        « .  1       ,    .       .   2a         -      ,  ^.  .    ma 

Thus,  J  taken  twice,  is  y-;  and  taken  m  times,  is  -j-. 

Again,  when  two  quantities  are  to  be  multiplied  together, 
cither  may  be  made  the  multiplier  (Art.  67);  to  multiply  4  by  f. 

is  the  same  as  to  multiply  §  by  4.     Or,  to  multiply  m  by  r,  is  the 
same  as  to  multiply  j  by  m.     Hence,  the 

RULE, 

FOR    THE    MULTIPLICATION    OF    A    FRACTION    BY    AN    ENTIRE   QUANTITY, 
OR   OF    AN    ENTIRE    QUANTITY    BY    A    FRACTION. 

Multiply  the  numerator  hy  the  entire  quantity,  and  write  the  pro- 
duct over  the  denominator. 

Since  (See  Art.  125,)  dividing  the  denominator  of  a  fraction 
increases  the  value  of  the  fraction,  as  many  times  as  there  are 
units  in  the  divisor,  it  is  evident,  that  any  fraction  will  be  multi- 
plied by  an  entire  quantity,  if  the  denominator  of  the  fraction  be 
divided  by  the  entire  quantity.  Thus,  in  multiplying  |  by  2,  we 
may  divide  the  denominator  by  2,  and  the  result  will  be  f ,  which 
is  the  same  as  to  multiply  by  2,  and  reduce  the  resulting  fraction 
to  its  lowest  terms.  Hence,  in  multiplying  a  fraction  and  an  entire 
quantity  together,  we  should  always  divide  the  denominator  of  the  frac- 
tion hy  the  entire  quantity,  when  it  can  he  done  without  a  remainder. 

Remark  . — The  expression,  "  What  is  two  thirds  of  6  ?  "  has  the  same 
meaning,  as  "  AVhat  is  the  product  of  6  multiplied  by  §  ?  "  The  reason  of 
the  rule  for  the  multiplication  of  an  entire  quantity  by  a  fraction,  may  be 

shown  otherwise,  thus  :  one  third  of  a  is  -  ;  two  thirds  is  twice  as  much  as 

o 

one  third,  that  is,  two  thirds  of  a  is  _^.     Also,  -  of  a  is  -,  and  the  -  part 

3  n  n  n 

ma 
of  a  is    — . 
11 

Review. — 139.  How  do  you  multiply  a  fraction  by  an  entire  quantity, 
or  an  entire  quantity  by  a  fraction  ?  When  the  denominator  of  the  frac- 
tion is  a  multiple  of  the  entire  quantity,  what  is  the  shortest  method  of 
finding  their  product  ? 


MULTIPLICATION   OF   FRACTIONS.  101 


EXAMPLES. 

1.  Multiply  J—  hj  ad Ans.  — — . 

2.  Multiply  4^byxy Ans.  ^^^. 

3.  Multiply  ^  by  b-c Ans.  ^-. 

4.  Multiply  jQ-  by  5y Ans.  -^  . 

5.  Multiply  a-26  by  g;^ ^"^- -3^+^- 

6.  Multiply  a'-b'  by  ^'^.  .    .      Ans.  3a''=-^V'c-a^+aV^ 

7.  Multiply  -r^  by  a+c Ans.  —       '  — !—  . 

8.  Multiply  ^^-^  by  a-6 Ans.  -^^, 

9.  Multiply  jj^ —  by  a6 Ans. ■ . 

10.  Multiply  -,,#^^^3  by  2:^y-     •    •    .Ans.-|^^#-. 

11.  Multiply  3^-^^  by  x^+2/^ Ans.  g^J^. 

12.  Multiply  j^;^-^^  by  2(«-i).      .    .    .Ans.^^^. 

13.  Multiply  ^1^=^^  by  5(«-6)(e+<i).     Ans.  ^^t^^. 

14.  Multiply  -  by  c Ans.  — =a,  or  j. 

Hence,  we  see,  that  if  a  fraction  is  multiplied  by  a  quantity 
equal  to  its  denominator,  the  product  will  be  equal  to  the  numerator. 

15.  Multiply  — —  by  c+ J Ans.  a— &. 

m  '^ — ?2 

16.  Multiply  ^ — -^  by  2a;+5y Ans.  ?w'^— 7J^ 

Art.  140.— To  multiply  a  fraction  by  a  fraction. 

1.  Let  it  be  required  to  find  the  product  of  7^  multiplied  by  f . 

Since  f  is  the  same  as  2  multiplied  by  ;',  it  is  required  to  mul- 
tiply I  by  2,  and  take  \  of  the  product.  Now,  |  multiplied  by  2, 
is  equal  to  |,  and  \  of  f ,  is  equal  to  -f^  (since,  to  take  \  is  to 
divide  by  3,  and  any  fraction  is  divided,  by  multiplying  its  denom- 
inator, by  Art.  124.)     Hence,  the  product  of  J  and  |  is  f^. 


102  RAY'S    ALGEBRA,    PART  FIRST. 

In  the  same  manner,  if  it  were  required  to  multiply  -  by  — ,* 
since  — =wX-,  we  would  multiply  -  by  m,  and  take  -  of  the 

71f  71f  C  11/ 

product.     Thus,  -X»i= — ,  and  -  of = — .     Hence,  the 

^  c  c  n        c       nc 

RULE, 

FOR    THE    MULTIPLICATION    OF    A    FRACTION    BY   A    FRACTION. 

Multiply  the  numerators  together,  for  a  new  numerator;  and  the 
denominators  together,  for  a  new  denominator. 

Remarks.  —  1st  If  either  of  the  factors  is  a  mixed  quantity,  it  is 
best  to  reduce  it  to  an  improper  fraction,  before  commencing  the  operation. 

2d.  The  expression,  "  What  is  one  third  of  one  fourth,"  has  the  same 
meaning  as  "  What  is  the  product  of  i  multiplied  by  J."  Also,  the  expres- 
sion, "What  is  two  thirds  of  three  fourths,"  has  the  same  meaning  as 
*'  What  is  the  product  of  J  multiplied  by  §." 

3d.  When  the  numerators  and  denominators  have  common  factors,  it  is 
best  to  indicate  the  multiplication,  and  then  cancel  the  factors  common  to 
both  terms,  after  which,  the  remaining  terms  may  be  multiplied  together. 

Th       — -  V  —  —    ^X^Xfa  ___??c 
inus,  i^ij>^2ld~  ^X'SXSX1lbd~\ibd' 
Ako       5a   s^,(f'-\-b^      5a{a+b)      _      5 
'   a;'—b'^  2a      2a[a+b){a—h)~'2{a—b)' 

EXAMPLES. 

1    TIT  u-  1    3a        5x  15aa: 

1.  Multiply^  by  -g Ans.  -^. 

2.MuUipl/g?byf5 Ans-gL^. 

o    n/r  ^^'  .    2a  ,     4a  .        Sa^ 

3.  Multiply -iT-  by -^ Ans. -ip-^. 

4.  Multiply  ~  by  I Ans.  |. 

5.  Multiply  li2+5l  by  -^ Ans.  Cx. 

6.  Multiply — ^ —  Ry -y- Ans. — ^^ . 

Review. — 140.  How  do  you  multiply  one  fraction  by  another  ?  Explain 
the  reason  of  the  rule,  by  analyzing  an  example.  When  one  of  the  factors 
is  a  mixed  quantity,  what  ought  to  be  done  ?  What  is  the  meaning  of  the 
expression,  "  What  is  one  third  of  one  fourth  ?  "  How  may  the  work  bo 
shortened,  when  the  numerator  and  denominator  have  common  factors? 


MULTIPLICATION   OF   FRACTIONS.  103 

7.MuUip„±^^-\,^-J-^ An.ta). 

8.  Multiply -^^  by  5^^- Ans.  1. 

9.MultipIy^^by^,. An.-^, 

10.  Multiply  ?=^  by  ;^ A„s.  ^. 

11.  Multiply  — — , ,  and ,  together.    .    .    .     Ans.  -. 

a-j-x       x^  a~x      ^  X 

/>»2    I_^y2     /y,2 «/2 

12.  Multiply  — ^-,  — -^-,  and  a  together.     .    Ans.  a{x^+f). 

X    y     x-j-y 

13.  Multiply    ^    ,  — — ~j,  and  a+6  together.     .    .    .     Ans.  1. 

14.  Multiply  "P^l  by  ^ Ans.  ---^?^^. 

^  ''  x^—i/        x-\-y  x^-^2xy^y^ 

15.  Multiply  2=*  by  i^^ '  Ans.  ?5+?. 

16.  Multiply  .+3^  by  ^- Ans.  M. 

CASE    VII. 

TO    DIVIDE    ONE    FRACTIONAL   QUANTITY  BY    ANOTHER. 

Art.  141. — To  divide  a  fraction  by  an  entire  quantity. 

It  has  been  shown,  in  Art.  123  and  Art.  124,  that  a  fraction  is 
divided  by  an  entire  quantity,  by  dividing  its  numerator,  or  multi- 
plying its  denominator.     Thus,  f  divided  by  2,  or  h  of  |,  is  §. 

3«  -,..■,-,  V    o        ,      „  3a   .    a      »»«  J-  -J  J  .  In 

~r  divided  by  o,  or  4  of  -^,  is  v- divided  by  m,  or  —  of 

ma    .     a 

,  is  -. 

n         n 

Or,  by  multiplying  the  denominator ;  ^  divided  by  2,  is  equal  to 

•/o.  since  the  number  of  parts  in  the  numerator  is  the  same,  but 

only  half  as  large  as  before,  jo  being  the  half  of  i.      Hence,  the 

RULE, 

FOR    DIVIDING    A    FRACTION    BY   AN    ENTIRE    QUANTITY. 

Divide  the  numerator  by  the  divisor,  if  it  can  be  done  without  a 
remainder;  if  not,  multiply  the  denominator  by  the  entire  quantity, 
and  write  the  numerator  over  the  residt. 

Note. — If  the  numerator  of  the  fraction  and  the  entire  quantity,  cqn- 
tain  common  factors,  it  is  best  to  indicate  the  operation,  and  cancel  the 
common  factors;  the  result  found  thus  will  bo  in  its  lowest  terms. 


104  RAY'S    ALGEBRA,    PART    FIRST. 

The  preceding  rule  may  be  derived  in  another  manner,  thus: 
To  divide  a  number  by  2,  is  to  take  |  of  it,  or  to  multiply  it  by  -| ; 
to  divide  by  3,  is  to  take  ^  of  it,  or  to  multiply  it  by  J.     In  the 

Bame  manner,  to  divide  a  quantity  by  m,  is  to  take  —  of  it,  or  to 
multiply  it  by  — .  Hence,  to  divide  a  fraction  hy  an  entire  quan- 
tity, we  write  the  divisor  in  the  form  of  a  fraction  (thus,  m=^), 
and  invert  it,  and  then  proceed  as  in  midtiplication  of  fractions. 


EXAMPLES. 

1.  Divide -= — by  3a6 » Ans.  s^. 

In     ^  In 

2.  Divide  i^|\y5a^c ^^«- iH* 

3.  Divide     , ,    —  by  lacm^ Ans.  y\ —  • 

4.  Divide  -^ — n-  by  bb^d Ans.  -^t^ — -. 

-    ^.  .  -     d^-\-ah  .  .         a-\-h 

5.  Divide  K-;~7»-  '->y  « -^"s.  o-tt*-' 

d+2aj    •'  «3+2a; 

c^~\~cd  c 

6.  Divide  — = —  by  c-\-d Ans.  ^. 

7.  Divide  5!±^^+l!byx+y Ans.  4^. 

o.  Divide  ...  ,  »    by  a—b Ans.     ,,,  ,  ,^ — . 

26+3c    -^  26+3c 

9.  Divide by  3a+5 Ans. . 

c-g  ^  c—g 

10.  Divide  ?^-F^^  by  4x2+2a:-3 Ans.  ^^,. 

ab-Jrcd^        *'  a6+C(i- 

11.  Divide  i^-  by  6 Ans.  ^^j-. 

Sc    ''  36c 

3  3 

1^-  ^^^^^^^+^^^^^ ^"^-^6^6^^- 

13.  Divide  ?±^,-  by  a+6 Ans.  ^,^. 

Review. — 141.  How  do  you  divide  a  fraction  by  an  entire  quantity? 
Explain  the  reason  of  the  rule,  by  analyzing  an  example.  How  may  the 
work  be  abbreviated,  when  the  numerators  of  the  fraction  and  the  entire 
quantity  contain  common  factors  ? 


DIVISION   OP   FRACTIONS.  105 

14.  Divide?^  by  xy Ana.  ;r4£±|^-.. 

ic^  T\'A     3a+5c        o      o  A         3a+5c 

^^-^'^^^a^-Sr*^''^""'' ^°^-J5-^ 

17.  Divide -,,^^  by  a;+2, A-«- 1=|.- 

18.  Divide  -;r ^ — ^  by  a^-\-ax-\-x^.     .    .       Ans.  -r-^ — — i— — ^,. 

19.  Divide  — j— —  by  a+6c Ans.  ■^-^. 

20.  Divide — by  2a/b Ans.  ^rr- — r^. 

21.  Divide  — ^— —  by  am — an Ans.  —r-, — . 

6+c      *'  ab-\-ac 

22.  Divide  s-To-  by  a6+?;^ Ans.    ,,,  ,  o, — . 

2+3x    -^        '  2b-\-Sbx 

23.  Divide  ^^  by  x'—xy Ans.  ^. 

24.  Divide by  a'^+ab+b^ Ans.  — —. 

Art.  142.— To  divide  an  integral  or  fractional  quantity  by  a 
fraction. 

1.  How  often  is  |  contained  in  4,  or  what  is  the  quotient  of  4 
divided  by  |  ? 

4  is  equal  to  7=^/,  and  2  thirds  (f ),  is  contained  in  12  thirds 
(*/),  as  often  as  2  is  contained  in  12,  that  is,  6  times. 

m 

2.  How  often  is  —  contained  in  a? 

n 

a     na        ,  m  .  .      ,  .    na         „, 

a  is  eaual  to  t= — ,  find  —  is  contained  in  —  as  otten  as  m  is 
^  I      11  n  n 

contained  in  na,  that  is  —  times.      Or,  -  is  contained  in  a,  na 

times:  hence,  mX-,  or  —  is  contained  —  as  many  times,  that  is, 
'  n         n  wi 

7ia  ^. 

—  times. 

m 

3.  How  often  is  |  contained  in  |? 

Here,  3=7^5,  and  f =7^2,  and  8  twelfths  (7^0)  is  contained  in  9 
twelfths  (7H7),  as  often  as  8  is  contained  in  9,  that  is,  1=1  g  times. 


106  RAY'S   ALGEBRA,    PART  FIRST. 


4.  How  often  is  —  contained  in  -? 
n  c 

Reducing  these  fractions  to  a  common  denominator,  — = —  ,  and 

n      nc 

a    na  mc  .  .      .  .    na         „  .  .      -,   . 

-= — ;  now,  —  IS  contamed  m  —  as  often  as  mc  is  contained  in 
c     nc  nc  nc 

na,  that  is,  —  times.     This  is  the  same  result  as  that  produced  by 

multiplying  -  by  —  inverted,  that  is  -X — = —  • 
^  *'     *=  c    -^    n  c     m     mc 

An  examination  of  each  of  these  examples,  will  show  that  the 
process  consists  in  reducing  the  quantities  to  a  common  denomina- 
tor, and  then  dividing  the  numerator  of  the  dividend,  by  the  nume- 
rator  of  the  divisor.  But,  as  the  common  denominator  of  the 
fraction  is  not  used  in  performing  the  division,  the  result  will  be 
the  same  as  if  we  invert  the  divisor,  and  proceed  as  in  multiplica- 
tion.    Hence,  the 

RULE, 

FOR  DIVIDING  AN  INTEGRAL  OR  FRACTIONAL  QUANTITY  BY  A  FRACTION. 

Reduce  both  dividend  and  divisor  to  the  form  of  a  fraction;  then 
invert  the  terms  of  the  divisor,  and  multiply  the  numerators  together 
for  a  7iew  numerator,  and  the  denominators  together  for  a  new 
denominator. 

Note.  — After  inverting  the  divisor,  the  work  may  be  abbreviated,  by 
canceling  all  the  factors  common  to  both  terms  of  the  result. 


EXAMPLES. 

1.  Divide  4  by  ,-r Ans. — . 

-^  3  \  a 

2.  Divide  4  by  - Ans.  -^. 

^   a  3 

3.  Divide  a  by  ^ Ans.  4a. 

4.  Divide  ab"^  by  -^  - Ans.  -^-. 

^    he  2 

5.  Divide  a^-6' by  ^1 Ans.  ?%:*).. 

■^        Sa  2 

6.  Divide  5  by  ?^ Ans.  j^-. 

o         z  oc 

Review.  —  142.  How  do  you  divide  an  integral  or  fractional  quantity 
by  a  fraction  ?  Explain  the  reason  of  this  rule,  by  analyzing  an  example. 
When,  and  how,  can  the  work  be  abbreviated  ? 


DIVISION  OF  FRACTIONS.  107 

fy    ^.  .,    3a  ,     X  .9a 

7.  Divide  —  by  o Ans.  -:r. 

8.  Divide  — ^  ^Y  -r Ans.  -. 

cd     -^   d  c 

9.  Divide  4'^  by  f- Ans.  |*5. 

Sa     *'   2b  Say 

10.  Divide  -^-  by  -o- Ans.  ytx- 

n    Tk«  •  J    3^^^  r-    3aa;^  .        2a 

11.  Divide -^;3— by -^-j- Ans. — . 

7       "^    14  X 

1 6ax       4ic 

12.  Divide  — ^  ^^15 -^^s-  1^^- 

13.  Divide  ?fny?5? Ans.|. 

14.  Divlde^by^ Ans.  ^i^Zl^). 

5        ^     a  5 

15.  Divide  —^  by -^— Ans.  — „— • 

16.  Divide /— L£-  by  -r-^ Ans. . 

ah  ''     be  a 

'7W  77  '777    I    'W 

17.  Divide  — -^ —  by  —5-- Ans.  2m— 2w. 

18.  Divide  --:, — =-  by =- Ans.    „  ,  ^    ,  j. 

a^~\     '  a—\  a2+2a+l 

iQ    T^    -^    4a+12  -     3a+9  .       86 

19.  Divide  — g —   by  ~j^f Ans.  ^. 

on    Ti-  'A    2^+3  -     lOz+15  .       x-y 

20.  Divide  — r —  by — 7-^— .r Ans.  — ^. 

x-[-y     ''     x^ — y''  5 

21.  Divide  g  by  ^j^-j; Ans.  1. 

22.  Divide  5(2!=.--!>  by  2(^^1 K.J-^^^±^. 

X  •'      a— X  Zx 

23.  Divide  -^— -^  by  — — Ans.  -r, — ^. 

Art.  143.— To  reduce  a  complex  fraction  to  a  simple  one. 
This  may  be  regarded  as  a  case  of  division,  in  which  the  divi- 
dend and  the  divisor  are  either  fractions  or  mixed  quantities. 

Thus,  — ,  is  the  same  as  to  divide  2  J  by  3\, 
3^ 

,b 

a-\ —  , 

Also,  ,  is  the  same  as  to  divide  a-\—  by  w-f-  . 

m-\ — 
r 


108  RAY'S   ALGEBRA,    PART   FIRST. 


(«+!)-(-+:-)= 


ac+&  ,  mr-\-7i ac-{-b        r     acr-\-hr 

c      '       r  c        mr-\-n    cmr-\-cn 

In  the  same  manner,  let  the  following  examples  be  solved. 
a 

1.  Reduce — to  a  simple  fraction Ans.  — -. 

c  ^  be 

d 

3^ 
2  ,       21 

2.  Reduce  —  to  a  simple  fraction Ans.  ^. 

3 

r 

3.  Reduce  —  to  a  simple  fraction Ans. . 

T  vn 

4.  Reduce  —  to  a  simple  fraction Ans.  — . 

n 

5.  Reduce to  a  simple  fraction Ans. . 

m  cm 

6.  Reduce  =-  to  a  simple  fraction Ans.  — — ^. 

,  1  ^  ac+1 

A  complex  fraction  may  also  be  reduced  to  a  simple  one,  by 
multiplying  both  terms  by  the  least  common  multiple  of  the  denom- 
inators of  the  fractional  parts   of  each  term.      Thus,  we  may 

41 
reduce  —^  to  a  simple  fraction,  by  multiplying  both  terms  by  6, 
Do 

the  least  common  multiple  of  2  and  3;  the  result  is  ||.  In  some 
cases  this  is  a  shorter  method,  than  by  division.  Either  method 
may  be  used. 

Art.  144. — Resolution  of  fractions  into  series. 

An  injinite  series  consists  of  an  unlimited  number  of  terms, 
which  observe  the  same  law. 

The  law  of  a  series  is  a  relation  existing  between  its  terms,  so 
that  when  some  of  them  are  known,  the  succeeding  terms  may  be 
easily  derived. 

Review. — 143.  How  do  you  reduce  a  complex  fraction  to  a  simple  one, 
by  division  ?     How,  by  multiplication  ? 


RESOLUTION    OF    FRACTIONS   INTO    SERIES.      109 


Thus,  in  the  infinite  series,  1 — ax+aV— a'x'+aV,  &c.,  any 
term  may  be  found,  by  multiplying  the  preceding  term  by — ax. 

Any  proper  algebraic  fraction,  whose  denominator  is  a  polyno- 
mial, can,  by  division,  be  resolved  into  an  infinite  series;  for,  the 
numerator  is  a  dividend,  and  the  denominator  a  divisor,  so  related 
to  each  other,  that  the  process  of  division  never  can  terminate, 
and  the  quotient  will,  therefore,  be  an  infinite  series.  After  a  few 
of  the  terms  of  the  quotient  are  found,  the  law  of  the  series  will, 
in  general,  be  easily  seen,  so  that  the  succeeding  terms  may  be 
found  without  continuing  the  division. 

EXAMPLES. 

1.  Convert  the  fraction  -: into  an  infinite  series. 

1 — X 

1  IJ-x 

I X        \-\-x-{-x^-\-x^-\-,  &c.         The  laAv  of  this  series  evidently 

1^  is,  that  each  term   is  equal  to  the 

_|__^ ^2  preceding  term,  multiplied  by  -{-x. 

+x^ 


From  this,  it  appears,  that  the  fraction  ^j ,  is  equal  to  the  infi- 
nite series,  l-\-x~{-x^-\-x^+x*-i-,  &c. 

In  a  similar  manner,  let  each  of  the  following  fractions  be 
resolved  into  an  infinite  series,  by  division. 

2.    T- — =1 — x4-x^ — xi^-\-x* — ,  &c.,  to  infinity. 
l+x 

a — X  a      a^     a^ 

4.  l^=l-^2x+2x'+2x^+,  &c. 
1 — X 

5.  ^=1— 2x+2x2— 2x3+,  &c. 

x+1  X        X-     x-^ 

R  E  V I E  w. — 144.  What  is  an  infinite  series  ?  What  is  the  law  of  a  series  ? 
Give  an  example.  Why  can  any  proper  algebraic  fraction,  whose  denom- 
inator is  a  polynomial,  be  resolved  into  an  infinite  series,  by  division  ? 


110  RAY'S   ALGEBRA,    PART   FIRST. 


CHAPTER  lY. 

EQUATIONS    OF    THE    FIRST    DEGREE. 

DEFINITIONS  AND  ELEMENTARY  PRINCIPLES. 

Art.  145. — The  most  useful  part  of  Algebra,  is  that  which 
relates  to  the  solution  of  problems.  This  is  performed  by  means 
of  equations. 

An  equation  is  an  Algebraic  expression,  stating  the  equality 
between  two  quantities. 

Thus,  X — 3=4,  is  an  equation,  stating,  that  if  3  be  subtracted 
from  X,  the  remainder  will  be  equal  to  4. 

Art.  146.— Every  question  is  composed  of  two  parts,  separated 
from  each  other  by  the  sign  of  equality.  The  quantity  on  the  left 
of  the  sign  of  equality,  is  called  the  first  member,  or  side  of  the 
equation.  The  quantity  on  the  right,  is  called  the  second  memhery 
or  side.  The  members  or  quantities  are  each  composed  of  one  or 
more  terms. 

Art.  I4'7. — There  are  generally  two  classes  of  quantities  in  an 
equation,  the  known  and  the  unknown.  The  known  quantities  are 
represented  either  by  numbers,  or  the  first  letters  of  the  alphabet, 
as  a,  b,  c,  &c.;  and  the  unknown  quantities  by  the  last  letters  of 
the  alphabet,  as  x,  i/,  z,  &c. 

Art.  148. — Equations  are  divided  into  degrees,  called  first, 
second,  third,  and  so  on.  The  degree  of  an  equation,  depends  on 
the  highest  power  of  the  unknown  quantity  which  it  contains. 

An  equation  which  contains  no  power  of  the  unknown  quantity 
higher  than  the  first,  is  called  an  equation  of  the  first  degree. 

Thus,  2x'-j-5=^9,  and  ax-\-b=^c,  are  equations  of  the  first  degree. 
Equations  of  the  first  degree  are  usually  called  Simple  Equations. 

An  equation  in  which  the  highest  power  of  the  unknown  quan- 
tity is  of  the  second  degree,  that  is,  a  square,  is  called  an  equation 
of  the  second  degree,  or  a  quadraiic  equation. 

R  E  V I  E  w. — 145.  What  is  an  equation  ?  Give  an  example.  146.  Of  how- 
many  parts  is  every  equation  composed?  How  are  they  separated?  What 
is  the  quantity  on  the  left  of  the  sign  of  equality  called  ?  On  the  right  ?  Of 
what  is  each  member  composed?  147.  How  many  classes  of  quantities  are 
there  in  an  equation?  How  are  the  known  quantities  represented?  How 
are  the  unknown  quantities  represented  ?  148.  How  are  equations  divided  ? 
On  what  does  the  degree  of  an  equation  depend  ?  What  is  an  equation  of 
the  first  degree?  Give  an  example.  What  are  equations  of  the  first  degree 
usually  called  ?  What  is  an  equation  of  the  second  degree  ?  Give  an  exam- 
ple.    What  are  equations  of  the  second  degree  usually  called. 


SIMPLE   EQUATIONS.  Ill 

Thus,  4x'^ — 7=29,  and  ax'^-\-bx=c,  are  equations  of  the  second 
degree. 

In  a  similar  manner,  we  have  equations  of  the  ihi7'd  degree, 
fourth  degree,  &c.;  the  degree  of  the  equation  being  always  the 
same  as  the  highest  power  of  the  unknown  quantity  which  it 
contains. 

When  any  equation  contains  more  than  one  unknown  quantity, 
its  degree  is  equal  to  the  greatest  sum  of  the  exponents  of  the 
unknown  quantities  in  any  of  its  terms. 

Thus,  xy-\-ax-\-bi/=^c,  is  an  equation  of  the  second  degree. 
xhj-^-x^  -\-cx  =^a,  is  an  equation  of  the  third  degree. 

Art.  149. — An  identical  equation,  is  one  in  which  the  two  mem- 
bers are  identical ;  or,  one  in  which  one  of  the  members  is  the 
result  of  the  operations  indicated  in  the  other. 

Thus,  2x—l=2x—\,  5x+3a:=8x-,  and  (a:+2)(x— 2)=x2— 4, 
are  identical  equations. 

Equations  are  also  distinguished  as  numerical  and  literal.  A 
numerical  equation  is  one  in  which  all  the  known  quantities  are 
expressed  by  numbers. 

Thus,  x^+2a:=3x+7,  is  a  numerical  equation. 

A  literal  equation  is  one  in  which  the  known  quantities  are  rep- 
resented by  letters,  or  by  letters  and  numbers. 

Thus,  ax — h=cx-x-d,  and  aa;^+&'^=2x — 5,  are  literal  equations. 

Art.  150. — Every  equation  is  to  be  regarded  as  the  statement, 
in  algebraic  language,  of  a  particular  question. 

Thus,  X — 3=4,  may  be  regarded  as  the  statement  of  the  follow- 
ing question :  To  lind  a  number,  from  which,  if  3  be  subtracted, 
the  remainder  will  be  equal  to  4. 

If  we  add  3  to  each  member,  we  shall  have  x — 3+3=4+3,  or 
a:=7. 

An  equation  is  said  to  be  verified,  when  the  value  of  the  unknown 
quantity  being  substituted  for  it,  the  two  members  are  rendered 
equal  to  each  other. 

Thus,  in  the  equation  x — 3=^4,  if  7,  the  value  of  x,  be  substi- 
tuted instead  of  it,  we  have  7 — 3=4,   or,  4=4. 

To  solve  an  equation,  is  to  find  the  value  of  the  unknown  quajitity ; 
or,  to  find  a  number,  which  being  substituted  for  the  unknown 
quantity,  will  render  the  two  members  identical. 

Keview. — 148.  When  an  equation  contains  more  than  one  unknown 
quantity,  to  what  is  its  degree  equal  ?  Give  an  example.  149.  What  is  an 
identical  equation  ?  Give  examples.  What  is  a  numerical  equation  ?  Give 
an  example.  What  is  a  literal  equation?  Give  an  example.  150.  How  is 
every  equation  to  be  regarded  ?  Give  an  example.  When  is  an  equation 
said  to  be  verified?     What  do  you  understand,  by  solving  an  equation  ? 


112  RAT'S   ALGEBRA,    PART    FIRST. 


Art.  151. — The  value  of  the  unknown  quantity  in  any  equa- 
tion, is  called  the  root  of  that  equation. 

SIMPLE  EftUATIO^fS,  COi\TAIi\Ii\  G  BUT  OIVE   LXRIVOWIV 
QUAIVTITY. 

Art.  152. — The  operations  that  we  employ,  to  find  the  value 
of  the  unknown  quantity  in  any  equation,  are  founded  on  this 
evident  principle:  If  ice  jjerform  exactly  the  same  operation  on  two 
equal  quantities,  the  results  will  be  equal.  This  principle,  or  axiom, 
may  be  otherwise  stated,  as  follows : 

1 .  If,to  two  equal  quantities,  the  same  quantity  he  added,  the  sums 
will  he  equal. 

2.  If,  from  two  equal  quantities,  the  same  quantity  he  suhtracted, 
the  remainders  ivill  be  equal. 

3.  If  two  equal  quantities  he  multiplied  by  the  same  quantity,  the 
products  will  be  equal. 

4.  If  two  equal  quantities  he  divided  by  the  same  quantity,  the 
quotients  loill  be  equal. 

5.  If  two  equal  quantities  he  raised  to  the  same  power,  the  results 
will  he  equal. 

6.  If  the  same  root  of  tivo  equal  quantities  be  extracted,  the  results 
will  be  equal. 

R  E  M  A  R  K . — An  axiom  is  a  self-evident  truth.  The  preceding  axioms 
are  the  foundation  of  a  large  portion  of  the  reasoning  in  mathematics. 

Art.  153. — There  are  two  operations  of  frequent  use  in  the 
solution  of  equations.  These  are,  first,  to  clear  an  equation  of  frac- 
tions;  and,  second,  to  transpose  the  terms,  in  order  to  find  the  value 
of  the  unhiown  quantity.  These  are  named  in  the  order  in  which 
they  are  generally  used,  in  the  solution  of  an  equation ;  we  shall, 
however,  first  consider  the  subject  of 

TRANSPOSITION. 

Suppose  we  have  the  equation  2x — 3=x4-5. 

Since,  by  the  preceding  principle,  the  equality  will  not  be 
afi"ected,  by  adding  the  same  quantity  to  both  members ;  or,  by 
subtracting  the  same  quantity  from  both  members ;  if  we  add  3 
to  each  member,  we  have  2x — 3+3=a;-f  5-|-3. 

If  we  subtract  x  from  each  member,  we  have 
2a;— x-3+3=a:— a:+5+3. 

Review.  — 151.  "What  is  the  root  of  an  equation?  152.  Upon  what 
principle  are  the  operations  founded,  that  are  used  in  solving  an  equation? 
What  are  the  axioms  Avhich  this  principle  embraces  ?  153.  What  two  opera- 
tions are  frequently  used,  in  the  solution  of  equations  ? 


SIMPLE   EQUATIONS.  113 

But,  — 3+^  cancel  each  other;  so,  also,  do  x — x;  omitting 
these,  we  have  2x — a-=:5+3. 

Now,  the  result  is  the  same  as  if  we  had  removed  the  terms  — 3 
and  -\-x,  to  the  opposite  members  of  the  equation,  and,  at  the  same 
time,  changed  their  signs. 

Again,  take  the  equation  ax-\-h^^c — dx. 

If  we  subtract  h  from  each  side,  and  add  dx  to  each  side,  we 
have  ax-\-dx=c—h. 

But,  this  result  is  also  the  same  as  if  we  had  removed  the  terms 
+6  and  — dx  to  the  opposite  members  of  the  equation,  and,  at  the 
same  time,  changed  their  signs.     Hence, 

Any  quantity  may  he  transposed  from  one  side  of  an  equation  to 
the  other,  if  at  the  same  time,  its  sign  be  changed. 

TO  CLEAR  AK    EQUATION  OF  FRACTIONS. 

Art.  154.— 1.  Let  it  be  required  to  clear  the  following  equa- 
tion of  fractions. 

2+3-^- 

Since  the  first  term  is  divided  by  2,  if  we  multiply  it  by  2,  the 
divisor  will  be  removed ;  but  if  we  multiply  the  first  term  by  2, 
Ave  must  multiply  all  the  other  terms  by  2,  in  order  to  preserve  the 
equality  of  the  members.     Multiplying  both  sides  by  2,  we  have 

.+1=10. 

Again,  since  the  second  term  is  divided  by  3,  if  we  multiply  it 
by  3,  the  divisor  will  be  removed;  but,  if  we  multiply  the  second 
term  by  3,  we  must  multiply  all  the  terms  by  3,  in  order  to  pre- 
serve the  equality  of  the  members.  Multiplying  both  sides  by  3, 
we  have  3a;-i-2a:=30. 

Instead  of  multiplying  first  by  2,  and  then  by  3,  it  is  plain  that 
we  might  have  multiplied  at  once,  by  2X3,  that  is,  by  the  product 
of  the  denominators. 

2.  Again,  let  it  be  required  to  clear  the  following  equation  of 
fractions. 

ab    be 
Since  the  first  term  is  divided  by  ab,  if  we  multiply  it  by  ab,  the 
divisor  will  be  removed ;  but,  if  we  multiply  the  first  term  by  ab, 
we  must  multiply  all  the  other  terms  by  ab,  in  order  to  preserve 
the  equality  of  the  members. 

R  E  V I E  w. — 154.  How  may  a  quantity  be  transposed  from  one  member  of 
an  equation  to  the  other?  Explain  the  principle  of  transposition  by  an 
example. 

10 


114  RAY'S    ALGEBRA,    PART   FIRST. 

Again,  since  the  second  term  is  divided  by  be,  if  we  multiply  it 
by  be,  the  divisor  will  be  removed;  but,  if  we  multiply  the  second 
term  by  be,  we  must  multiply  all  the  other  terms  by  be,  in  order  to 
preserve  the  equality  of  the  members.  Hence,  if  we  multiply  all 
the  terms  on  both  sides,  by  abXbe,  the  equation  will  be  cleared  of 
fractions. 

Instead,  however,  of  multiplying  every  term  by  ab\be,  it  is  evi- 
dent, that  if  each  term  be  multiplied  by  such  a  quantity  as  will 
contain  the  denominators  without  a  remainder,  that  all  the  denomi- 
nators will  be  removed.  This  quantity  is,  evidently,  the  least  com- 
mon multiple  of  the  denominators,  which,  in  this  case,  is  abc; 
then,  multiplying  both  sides  of  the  equation  by  abe,  we  have 
cx-\-ax=^abcd.     Hence,  the 

RULE, 

FOR   CLEARING    AN    EQUATION    OF    FRACTIONS. 

Find  the  least  common  multiple  of  all  the  denominators,  and  mul- 
iiply  each  term  of  the  equation  by  it. 

Clear  the  following  equations  of  fractions. 

1.  1+1=5 Ans.  3a;+2x=30. 

2.  ^  — 1=2 Ans.  4x-3x=24. 

3.  1+^+1=1 Ans,  20x+15x+12x=60. 

o     4     O 

4.  -.+5 — ^=To -^^s.  6x+3a:— 4a;=10. 

4  o      b     l/<2 

5.  ^  —  c+T7\=T7\ ^ns.  lOx — 6x+3x=21. 

5  5     lU     10 

6.  I -4=1+6 .    .  Ans.  3x-24=2x+3G. 

7.  ^-|==|+I| Ans.  15x-20=18+14x. 

8.  ^--1=*^— 4.     Ans.  lOx— 40-12=18-3x-120. 

o        5      10 

9.  ?^?-|.|=-.^+^.     Ans.  14a:-21+4x=.14x-42+10. 

10.  a;_E=?=5__^+i Ans.  6x-3x+9=30-2x-8. 

Z  «» 

11.  — I — cr=^b Ans.  2x+ax— 5a=2a6. 

a       li 

12.^+^=.? Ans.  48+8ax-24a=9x-27. 

X — o      o      4 

Review. — 154.  How  do  you  clear  an  equation  of  fractions?    Explain 

the  principle  by  an  example. 


SIMPLE    EQUATIONS.  115 

13.  ^H r=a. 

03—3    a—h 

Ans.  ax—hx-\-a—h-\-^x—cx—9^^c=a'^x—abx—^a'^-\-^ab. 

^^-  ^+^=^6^ '•    •  Ans.  ax-&x+ax+5a:=c. 

15.  £+^+^=^ Ans.  adf^-hcf-\-hde=hdfhx. 

SOLUTION  OF  EQUATIONS  OF  THE  FIRST  DEGREE,  CONTAINING 
ONLY  ONE  UNKNOWN  QUANTITY. 

Art.  155* — The  unknown  quantity  in  an  equation  may  be  com- 
bined with  the  known  quantities,  either  by  Addition,  Subtraction, 
Multiplication,  or  Division ;  or,  by  two  or  more  of  these  different 
methods. 

1.  Let  it  be  required  to  find  the  value  of  a:,  in  the  equation 

x+3=5, 
where  the  unknown  quantity  is  connected  by  addition. 
By  subtracting  8  from  each  side,  we  have  a:=5— 3=2. 

2.  Let  it  be  required  to  find  the  value  of  x,  in  the  equation 

X— 3=5, 
where  the  unknown  quantity  is  connected  by  subtraction. 
By  adding  3  to  each  side,  we  have  ic=:5+3=:8. 

3.  Let  it  be  required  to  find  the  value  of  x,  in  the  equation 

3a;=15, 
where  the  unknown  quantity  is  connected  by  multiplication.  • 

By  dividing  each  side  by  3,  we  have  x=-k^=:5. 

4.  Let  it  be  required  to  find  the  value  of  x,  in  the  equation 

-=2 
3       ' 

where  the  unknown  quantity  is  connected  by  division. 

By  midtiplying  each  side  by  3,  we  have  a;=2x3=:6. 

From  the  solution  of  these  examples,  we  see,  that  lohen  the 
wilcnoivn  quantitg  is  connected  by  addition,  it  is  to  be  separated  by 
subtraction.  When  it  is  connected  by  subtraction,  it  is  to  be  separa- 
ted by  addition.  When  it  is  connected  by  multiplication,  it  is  to  be 
separated  by  division.  And,  lohen  it  is  connected  by  division,  it  is 
to  be  separated  by  multiplication. 

5.  Find  the  value  of  x,  in  the  equation 

3a;— 3=x+5. 
By  transposing  the  terms  —3  and  x,  we  have 
3a;— a:=5+3, 
reducing,  2a;=:8, 
dividing  by  2,    .x=f =4. 


116  RAY'S   ALGEBRA,    PART    FIRST. 

Let  this  value  of  x  be  substituted  instead  of  a;,  in  the  original 
equation,  and,  if  it  is  the  true  value,  it  will  render  the  two  mem- 
bers equal  to  each  other. 

Original  equation,  ....  3x — 3=a;+5. 

Substituting  4  in  the  place  of  x,  it  becomes 

3X4—3=4+5,  or  9=9. 

The  operation  of  substituting  the  value  of  the  unknown  quan- 
tity instead  of  itself,  in  the  original  equation,  to  see  if  it  will  ren- 
der the  two  members  equal  to  each  other,  is  called  verification. 

The  preceding  equation  may  be  solved  thus: 

3a: — 3=a;-|-5.  By  adding  3  to  each  member,  we  have 

3x — 3+3=a;+5+3.     By  subtractings  from  each  member, 
we  have  3x— x — 3-f3^=a: — a;-|-54-3. 

But  — 3+3  cancel  each  other;  so,  also,  do  x  and  — x;  omitting 
these,  and  then  reducing,  we  have  2x=8. 

Dividing  each  member  by  2,  .x=|=:4. 

K  E  M  A  n  K. — The  pupil  will  perceive  that  th6  two  methods  of  solution  are 
the  same  in  principle.  In  the  first,  we  use  transposition,  to  remove  the 
known  quantity  from  the  left  member  to  the  right,  and  the  unknown  quan- 
tity from  the  right  member  to  the  left.  In  the  second,  the  same  thing  is 
done,  by  adding  equals  to  each  member,  and  subtracting  equals  from  each 
member — this  being  the  principle  on  which  transposition  is  founded.  It  is 
recommended  to  the  teacher,  to  use  the  latter  method  until  the  principle  is 
well  understood  by  the  pupil,  after  which  the  first  method  may  be  used 
exclusively. 

-J. 2  a;+2 

6.  Find  the  value  of  a;  in  the  equation  x — -:=4-| — — -. 

o  o         / 

Multiplying  both  sides  by  15,  the  least  common  multiple  of  tho 
denominators,  we  have  15a; — (5a; — 10)=60+3x+6. 
or,   15a;- 5a:+10  =60+3a:+6. 
by  transposition,   15a;—  5x — 3a;  =60+6  —10. 
reducing,     7a;^=56. 
dividing,        x=8. 

X  X 

7.  Find  the  value  of  x  in  the  equation  -  —d=.—\-c. 

^  h  a 

multiplying  both  sides  by  ah,  ax  — abd=rhx  ~\-ahc. 

transposing,  ax  — hx  =^ahc  -\-abd. 

separating  into  factors,    {a~b)x  =ab{c-{-d). 

,.  .,.       ,      ,       ,.  ab{c+d) 

dividmg  by  (a — o),      x= —     j     - 

R  E  V I E  w. — 155.  What  arc  the  methods  by  which  the  unknown  quantity 
in  an  equation  may  be  combined  with  known  quantities  ?  Give  examples. 
When  tho  unknown  quantity  is  connected  by  addition,  how  can  it  be  sepa- 
rated ?  When,  by  subtraction  ?  I}y  multiplication  ?  By  division  ?  What 
is  verification  ? 


SIMPLE   EQUATIONS.  117 

From  the  preceding  examples  and  illustrations,  we  derive  the 

RULE, 

FOR    THE    SOLUTION    OF    AN    EQUATION    OF    THE    FIRST    DEGREE. 

1.  If  necessary,  clear  the  equation  of  fractions  ;  perfoiin  all  the 
operations  indicated;  and  transpose  all  the  terms  containing  the 
unknown  quantity  to  one  side,  and  the  knoion  quantities  to  the  other. 

2.  Reduce  each  member  to  its  simplest  form,  and  divide  both  sides 
by  the  coefficient  of  the  unknown  quantity. 


EXAxMPLES    FOR    PRACTICE. 

Note. — Let  the  pupil  verify  the  value  of  the  unknown  quantity  in  each 
example. 

1.  3a:-5=2x+7.   . Ans.  a:=12. 

2.  3a:— 8=16— 5a: Ans.  a:=3. 

3.  5a:-7=3a:+15       Ans.  .t=:11. 

4.  3x— 25=— a:— 9 Ans.  x=4. 

5.  15-2x=6a:  -25 Ans.  a:=5. 

6.  5(a;+l)  +  6(x+2)=9(a:+3) Ans.  a;=5. 

7.  4(5x-3)— 64(3-x)-3(12a:-4)=90.     .    .    .   Ans.  x=Q. 

8.  10(a:-f-5)-f8(a;+4)=5(a:+13)  +  121 Ans.  a;=8. 

9.  ^-2=5-'^ Ans.  a=10. 

2  5 

10.  |-|+7=|-|+10i Ans.  a;=12. 

11.  x+|+^=18 Ans.a:=8. 

12.  |+|-|=-14 Ans.a:=24. 

13.  ^+|-|=2J Ans.a:=2. 

14.  ^_2=1_^ Ans.x=2. 

15.  ?^1_-^=10+^ Ans.x=14. 

16.  ^-5=?..x-2-5=i A„s.a=7. 

17.  ?^2_4-.^2^_7^2 ^„^^^2 

Revikw. — 155.  What  is  the  rule  for  the  solution  of  an  equation  of  the 
first  degree,  containing  one  unknown  quantity? 

4 


118  RAY'S   ALGEBRA,   PART  FIRST. 

18.  |x— |a:+18=g(4a;+l) Ans.  a;=20. 

19.  ^=1 Ans.a:=l. 

20.  2x-^=x+^? Ans.x=li. 

^,     3      x-2    5      x+S 

21.  ^ 3-=^ ^ Ans.  a;=ll. 

x+S      x—S     x—5      ^  A  lO 

22.  — 5 = — =-Tz 2 Ans.  x=13. 

4  5  2 

23.  t?_|_6_^=^+3 ,    .     Aiis.a:=ll. 

^.    ^       2x+ll      4x— 6         7— 8a:  ,  ^ 

24.  2x = r^-= ;= — Ans.  x=7. 

5  117 

f._     a:+7      _,     2a:+5  ,  10— 5a:  .  o 

25.  —iz 5|=^ 1 5 — Ans.  a;=8. 

o  /  o 

^    X     2(a;— 1)     7a:— 4     a:— 1  .  ^ 

^^'   8+-^--=-T5— er Ans.a:=2. 


27.  4a:-6=2a:— ^ Ans.  a;=^. 

28.  axArh^=^cx-\-d. Ans.  a:=— — . 

a—c 

29.  ax — 6x=(? — ex Ans.  x= 7. 

a+c — b 

30.  ax—bx=.c-\-dx — e Ans.  a;= -, ,. 

a — 0 — d 

31.  7+9a-5x=6x+5aa: Ans.  a:^H^^, . 

5aH-ll 

32.  6(a— 6x)+c(aa:— c)=6c Ans.  a;= — r- . 

6^ — ac 

33.  (a+Z>)(6_a:)  +  (a— 6)(a+x)-=cl  .    .    .     Ans.  a:^'^-^:^--. 

34.  — |-r=c Ans.  x= — — , . 

a    0  a-{-h 

35.  — =6c+- Ans.  x= — ; — . 

x  X  be 

36.  — I 1 — =1 Ans.  a:=a+7;-f  c. 

XXX  '        ' 

Q-r    ^  I  .     ^      7  A  ab{c-i-d) 

o7.    — \-c=y-~d Ans.  x= — ^— ^--. 

a  0  a — 6 


SIMPLE   EQUATIONS.  119 


XXX  abed 

38.      -\-r'Jr-=d Ans.  a:=-7- --j-. 

a     o    c  ab-{-ac-\-oc 

o\).   -=~— +1 ^ns.  a;=— —-- . 

a—b     a+b  2b 

.^    X  .  X      X     ^  .  abc 

40.  -+- =1 Ans.  a;= -. 

abc  ac-\-bc — ab 

41.  ^4-^4-^=2 Ans.  x=i{ab+ac+bc). 

42.  ^+^-^=0 Ans.x=-^-. 

X    c      e  cd — be 

.ty     x—a      X — b    b  .  d^ 

4d.  —z ■=- Ans.  x= -. 

b  a       a  a — b 

l—x  a  2a+l 

45.   — =a6+6H — Ans.  x^^—tl — . 

XX  b 

.^    a — b a-\-b  .  Sac — be 

X — c     x-\-2c *  26 

QVESTIOIVS   PRODUCING   SIMPLE   EQUATIOIVS,   COi\TAIXIIVG 
Oi\LY    OIVE    Ui\Kl\0\Vi\    QUAiXTITY. 

Art.  156* — The  solution  of  a  problem,  by  Algebra,  consists  of 
two  distinct  parts. 

1  St.  To  express  the  conditions  of  the  problem  in  Algebraic  lan- 
guage; that  is,  to  form  the  equation. 

2d.  To  solve  the  equation;  that  is,  to  find  the  value  of  the  unJaiown 
quantity. 

With  pupils,  the  most  difficult  part  of  the  operation  of  solving 
a  question,  is  to  form  the  equation,  by  the  solution  of  which  the 
value  of  the  unknown  quantity  is  to  be  found.  Sometimes,  the 
statement  of  the  question  furnishes  the  equation  directly ;  and, 
sometimes,  it  is  necessary,  from  the  conditions  given,  to  deduce 
others,  from  which  to  form  the  equation.  When  the  conditions 
furnish  the  equation  directly,  they  are  called  explicit  conditions. 
When  the  conditions  are  deduced  from  those  given  in  the  question, 
they  are  called  implied  conditions. 

It  is  impossible  to  give  a  precise  rule,  by  means  of  which  every 
question  may  be  readily  stated  in  the  form  of  an  equation.  The 
first  point,  is,  to  understand  fully  the  nature  of  the  question,  so  as 
to  be  able  to  prove  whether  any  proposed  answer  is  correct. 

Review. — 156.  Of  what  two  parts  does  the  solution  of  a  problem  by 
Algebra,  consist?  What  are  explicit  conditions?  What  are  implied 
conditions  ? 


120  RAY'S   ALGEBRA,    PART   FIRST. 

After  this,  the  equation,  by  the  solution  of  which  the  value  of 
the  unknown  quantity  is  to  be  found,  may  generally  be  formed  by 
the  following 

RULE. 

Denote  the  required  quantity,  hy  one  of  ike  Jinal  letters  of  the 
alphabet;  then,  hy  means  of  signs,  indicate  the  same  operations  that 
it  2coidd  be  necessary  to  make  on  the  answer,  to  verify  it. 

Remarks. —  1st.  In  solving  a  question,  it  is  necessary  to  understand 
the  principles  of  the  science  which  it  involves,  at  least  so  far  as  they  relate 
to  the  question  under  consideration.  Thus,  when  a  problem  embraces  the 
consideration  of  Ratio  or  Proportion,  in  order  to  solve  it,  the  pupil  must  bo 
familiar  Avith  these  subjects.  In  the  following  examples,  the  learner  is  sup- 
posed to  be  acquainted  with  Ratio  and  Proportion,  as  far  as  they  are  taught 
in  Arithmetic.     (See  Ratio  and  Proportion,  Ray's  Arithmetic,  Part  III.) 

2d.  The  operations  concerned  in  the  solution  of  an  equation,  involve  tho 
removal  of  coefficients,  the  removal  of  denominators,  and  the  transposition 
of  quantities.  The  first  six  of  the  following  examples,  and  also  those  from 
the  16th  to  the  44th  inclusive,  are  arranged  with  reference  to  these  operations. 

EXAMPLES. 

1.  There  are  two  numbers,  tho  second  of  which  is  three  times 
the  first,  and  their  sum  is  48  ;  what  are  the  numbers  ? 

Let  x=  the  first  number. 

Then,  by  the  first  condition,  Sx=  the  second. 

And,  by  the  second  condition,  x+3x--=48. 

Reducing,  4a:=48. 

Dividing  by  4,  a;=12,  the  smaller  number. 

Then,  3x^=36,  the  larger  number. 

Proof,  or  verification.     12-|-36==48. 

2.  A  father  said  to  his  son,  "  The  difference  of  our  ages  is  48 
years,  and  I  am  5  times  as  old  as  you."     What  were  their  ages? 

Let  a;==  the  son's  age. 

Then  5^=  the  father's  age. 

And  bx — ^a:=48. 

Reducing,  4x=48. 

Dividing,  a;=12,  the  son's  age. 

Then  bx=QO,  the  father's  age. 

Verification.     60—12=48,  the  difference  of  their  ages. 

3.  What  number  is  that,  to  which,  if  its  third  part  be  added, 
the  sum  will  be  16? 

Let  x=  the  required  number. 

Review. — 156.  By  what  general  rule,  may  the  equation  of  a  problem 
be  found  ? 


SIMPLE   EQUATIONS.  121 

Then  the  third  part  of  it  will  be  represented  by  .j. 

o 

And,  by  the  conditions  of  the  question,  we  have  the  equation 

x+|=16. 

Multiplying  it  by  3,  to  clear  it  of  fractions,  3a;+a;=48. 
Reducing,  4x=48. 
Dividing,  a:=12. 

Verification.  12+  '/=12+4=16;  which  shows  that  the  value 
found  is  correct,  since  it  satisfies  the  conditions  of  the  question. 

Note.  — The  pupil  should  verify  the  answer  in  every  example. 

4.  What  number  is  that,  which  being  increased  by  its  half,  and 
then  diminished  by  its  two  thirds,  the  remainder  will  be  equal 
to  105. 

Let  0.=  the  number. 

Then  the  one  half  will  be  represented  by  j,,  and  the  two  thirds 
by^. 

And,  by  the  question  a;+^  —  ^=105. 

Multiplying  by  6,  6x+3a;— 4a:=630. 
Reducing,  5x=630. 
Dividing,  a;=126.     Ans. 

When  the  numbers  contained  in  a  solution  are  large,  it  is  some- 
times better  to  indicate  the  multiplication,  than  to  perform  it. 
The  preceding  solution  may  be  given  thus: 

6x+3a;-4x=105X6 
5a:=105X6 
x=  21X6=126. 

5.  It  is  required  to  divide  a  line  25  inches  long,  into  two  parts, 
60  that  the  greater  shall  be  3  inches  longer  than  the  less. 

Let  x=^  the  length  of  the  smaller  part. 
-   Then  aj+3=  the  greater  part. 
And  by  the  question,  a;+a:-f  3=25. 
Reducing,  2xH-3=25. 
Transposing  3,  2x=25— 3=22. 
Dividing,  ic=ll,  the  smaller  part. 
And  ic+3=14,  the  greater  part. 

6.  It  is  required  to  divide  68  dollars  between  A,  B,  and  C,  so  that 
B  shall  have  5  dollars  more  than  A,  and  C  7  dollars  more  than  B. 

11 


122  RAY'S   ALGEBRA,    PART   FIRST. 


Let  a;=:  A's  share. 
Then  x-\-5=  B's  share. 

And  a:+12=  C's  share.     Then,  by  the  terms  of  the  question^ 
we  have  a:+(x+5)+(a:+12)=68. 
Reducing,  3x+ 17=68. 
Transposing,  3x=68- 17=51. 
Dividing,  x=17,  A's  share. 
x-{-5  =22,  B's  share. 
a:+ 12=29,  C's  share. 

7.  AVhat  number  is  that,  which  being  added  to  its  third  part, 
the  sum  will  be  equal  to  its  half  added  to  10. 

Let  X  represent  the  number. 

Then,  the  number,  with  its  third  part,  is  represented  by  x-\-^ ; 

X  " 

and  its  half,  added  to  10,  is  expressed  by  ^+10.     By  the  condi- 
tions of  the  question,  these  are  equal;  that  is,  x-\-^=^-{-10. 

Multiplying  by  6,  6a;4-2a;=3ir  -  60. 

Reducing  and  transposing,     8a; — Sx  -  60. 

5x=60. 
Dividing,     x=12. 
Verification.     12+V^=*/  +  10.     Or  16=16,  according  to  the 
conditions. 

Hereafter,  we  shall,  in  general,  omit  the  terms,  transposing, 
dividing,  &c.,  as  the  various  steps  of  the  solution  will  be  evident 
by  inspection. 

8.  A  cistern  was  found  to  be  one  third  full  of  water,  and  after 
emptying  into  it  17  barrels  more,  it  was  found  to  be  half  full; 
what  number  of  barrels  will  it  contain  when  full  ? 

Let  a:=  the  number  of  barrels  the  cistern  will  contain. 

Then  |+17=|. 

2a:+102=3a: 
102=x 

Or,  by  first  transposing  3a;  and  102,  we  have  — a;= — 102  ; 
multiplying  both  sides  by  — 1,  we  have  x=102. 

The  unknown  quantity,  when  its  value  is  found,  is  generally 
made  to  stand  on  the  left  side  of  the  sign  of  equality  ;  it  is  not 
material,  however,  which  side  it  occupies,  since,  by  transposition, 
it  can  be  readily  removed  to  the  other.  In  effecting  the  transpo- 
sition of  102=x,  80  as  to  bring  the  x  on  the  left  side,  we  have 
made  it  to  consist  of  two  steps ;  it  is,  however,  generally  made  in 
one;  the  transposition,  and  multiplying  by  —1,  being  both  made 
in  one  line  at  the  same  time. 


SIMPLE   EQUATIONS.  123 

Note. — Multiplying  by — 1  is  the  same  as  changing  all  the  signs  of 
both  members  of  the  equation. 

9.  A  cistern  is  supplied  with  water,  by  two  pipes ;  the  less  alone 
can  fill  it  in  40  minutes,  and  the  greater  in  30  minutes  ;  in  what 
time  will  they  fill  it,  both  running  at  once  ? 

Let  x=^  the  number  of  min.  in  which  both  together  can  fill  it. 

Then  -=  the  part  which  both  can  fill  in  1  minute. 

Since  the  less  can  fill  it  in  40  minutes,  it  fills  4*^  of  it  in  1  min- 
ute. Since  the  greater  can  fill  it  in  30  minutes,  it  fills  ^^^  ^^  i*  ^^ 
1  minute.     Hence,  the  part  of  the  cistern  which  both  can  fill  in 

1  minute,  is  represented  by  TT^+on'  ^^^  ^^^o,  by  -. 

Hence,  3^+^=^. 

Multiply  both  sides  by  120a:,  and  we  have  3a:+4x=120. 

7a:=120. 
a:=J-|^=174  min. 

10.  A  laborer,  A,  can  perform  a  piece  of  work  in  5  days,  B  can 
do  the  same  in  6  days,  and  C  in  8  days ;  in  what  time  can  the 
three  together  perform  the  same  work  ? 

Let  ic=  the  number  of  days  in  which  all  three  can  do  it. 

Then  -=  the  part  which  all  can  do  in  1  day. 

If  A  can  do  it  in  5  days,  he  does  i  of  it  in  1  day. 

IfB     "        "       6    "  "J     " 

IfC     "        "       8    *'  "        I     " 

Hence,  the  part  of  the  work  done  by  A,  B,  and  C  in  1  day,  is 

represented  by  F+r.+5»  ^^^  ^l8<^>  by  ~  • 

Hence,  g+g+g—. 

Or,  24x+20a;+15a;=120. 
59x=120 

^=  W=2/^  days. 

11.  How  many  pounds  of  sugar  at  5  cents,  and  at  9  cents  per 
pound,  must  be  mixed,  to  make  a  box  of  100  pounds,  at  6  cents 
per  pound. 

Let  x=  the  number  of  pounds  at  5  cents. 
Then  100 — x=  the  number  of  pounds  at  9  cents. 
Also,  5ic=  the  value  of  the  former. 
And  9(100 — x)=  the  value  of  the  latter. 
And  600=  the  value  of  the  mixture. 


J  24  RAY'S   ALGEBRA  J    PART   FIRST. 

But  the  value  of  the  two  kinds  must  be  equal  to  the  value  of 
the  mixture. 
Therefore,  5a;+9(100— a:)=600 
5x+900-9x=600 
— 4x=-300 

x=^lb,  the  number  of  pounds  at  5  cents. 
100— x=25,      "         "         "        "        9  cents. 

12.  A  laborer  was  engaged  for  30  days.  For  each  day  that  he 
worked,  he  received  25  cents  and  his  boarding;  and,  for  each  day 
that  he  was  idle,  he  paid^O  cents  for  his  boarding.  At  the  expi- 
ration of  the  time,  he  received  3  dollars ;  how  many  days  did  he 
work,  and  how  many  days  was  he  idle  ? 

Let  a;=  the  number  of  days  he  worked. 
Then  30 — x=  the  number  of  days  he  was  idle. 
Also  2ox^=  wages  due  for  work. 

And  20(30—0:)—  the  amount  to  be  deducted  for  boarding. 
Therefore,  25x-20(30-x)=300 
25x-600+20a;  =300 
45x==:900 

a;=20=  the  number  of  days  he  worked. 
30 — a:=10=  the  number  of  days  he  was  idle. 
Proof.     25X20=500  cents,  =  wages. 

20X10=200  cents,  =  boarding. 

300  cents,  =  the  remainder. 
In  solving  this  example,  we  reduce  the  3  dollars  to  cents,  in 
order  that  the  quantities  on  both  sides  of  the  equation  may  be  of 
the  same  denomination.  For,  as  we  can  only  add  or  subtract  num- 
bers of  the  same  denomination,  it  is  evident,  that  we  can  only 
compare  quantities  of  the  same  name.  Hence,  all  the  quantities, 
in  both  members  of  an  equation,  must  be  of  the  same  denomination. 

13.  A  hare  is  50  leaps  before  a  greyhound,  and  takes  4  leaps 
to  the  greyhound's  3  ;  but  2  of  the  greyhound's  leaps  are  equal  to 
3  of  the  hare's ;  how  many  leaps  must  the  greyhound  take,  to 
catch  the  hare? 

Let  X  be  the  number  of  leaps  taken  by  the  hound.  Then,  since 
the  hare  takes  4  leaps  while  the-  hound  takes  3,  the  number  of 
leaps  taken  by  the  hare,  after  the  starting  of  the  hound,  will  be 

4x 

-H-  ;  and  the   whole  number  of  leaps  taken  by  the  hare,  will  be 
*> 

4x 

K-+50,  which  is  equal,  in  extent,  to  the  x  leaps  run  by  the  hound. 

o 

Now,  if  the  length  of  the  leaps  taken  by  each  were  equal,  we 

4iC 

might  put  X  equal  to  -h^+^O;  hut,  by  the  question,  2  leaps  of  the 


SIMPLE   EQUATIONS.  125 


hound  are  equal  to  3  of  the  hare's,  that  is,  1  leap  of  the  hound  is 
equal  to  |  leaps  of  the  hare  ;  hence,  x  leaps  of  the  hound  are 

equal  to  -^  leaps  of  the  hare ;  and  we  have  the  equation 

9a.'=8a;+300 
a:=:300,  leaps  taken  by  the  greyhound. 

14.  The  hour  and  minute  hands  of  a  watch  are  exactly  together 
between  8  and  9  o'clock ;  required  the  time. 

Let  the  number  of  minutes  more  than  40,  be  denoted  by  x;  that 
is,  let  x=:  the  minutes  from  VIII  to  the  point  of  coincidence,  P ; 
then,  the  hour  hand  moves  from  YIII  to  the  point  P,  while  the 
minute  hand  moves  from  XII  to  the  same  point ;  or,  the  former 
moves  over  x  minutes,  while  the  latter  moves  over  40+ic  minutes; 
but  the  minute  hand  moves  12  times  as  fast  as  the  hour  hand. 
Therefore,  12a;=404-a; 
llx=40 

a:=|fi  minutes  =3  minutes,  38fy  seconds. 

Hence,  the  required  time  is  43  minutes,  38yj  seconds  after  8 
o'clock. 

15.  A  person  spent  one  fourth  of  his  money,  and  then  received 
5  dollars.  He  next  spent  one  half  of  what  Jie  then  had,  and  found 
that  he  had  only  7  dollars  remaining ;  what  sum  had  he  at  first  ? 

Let  a:—  the  number  of  dollars  he  had  at  first.  Then,  after 
spending   one   fourth   of  that,  and   receiving   5  dollars,  he  had 

X  .  .  3x     „ 

X— j+S,  which  being  reduced,  is  equal  to  -j+5. 

He  now  spent  the  half  of  this  sum,  or  \  (   4;  +^  I  ~'ft'^"o* 

Therefore,  ^+5- (~+|]  =7; 
3x  ,  _     3x     h    ^ 

3.r     3x    ^  ,  5 

or,  4 --8=2+2; 

or,  6x—3x=- 16+20; 
3x=36 
x=12.     Ans. 
IG.  Divide  42  cents  between  A  and  B,  giving  to  B  twice  as 
many  as  to  A.  Ans.  A  14,  B  28. 

17.  Divide  the  number  48  into  three  parts,  so  that  the  second 
may  be  twice,  and  the  third  three  times  the  first. 

Ans.  8,  16,  and  24. 


i2G  RAY'S  ALGEBRA,    PART    FIRST. 

18.  Divide  the  number  60  into  3  parts,  so  that  the  second  may 
be  three  times  the  first,  and  the  third  double  the  second. 

Ans.  6,  18,  and  36. 

19.  A  boy  bought  an  equal  number  of  apples,  lemons,  and 
oranges,  for  56  cents ;  for  the  apples  he  gave  1  cent  a  piece,  for 
the  lemons  2  cents  a  piece,  and  for  the  oranges  5  cents  a  piece ; 
how  many  of  each  did  he  purchase  ?  Ans.  7. 

20.  A  boy  bought  5  apples  and  3  lemons,  for  22  cents  ;  he  gave 
as  much  for  1  lemon  as  for  2  apples ;  what  did  he  give  for  each  1 

Ans.  2  cents  for  an  apple,  and  4  cents  for  a  lemon. 

21.  The  age  of  A  is  double  that  of  B,  the  age  of  B  is  twice  that 
of  C,  and  the  sum  of  all  their  ages  is  98  years ;  what  is  the  age 
of  each  ?  Ans.  A  56  years,  B  28  years,  and  C  14  years. 

22.  Four  boys,  A,  B,  C,  and  D,  have,  between  them,  44  cents : 
of  which  A  has  a  certain  number,  B  has  three  times  as  many  as 
A,  C  as  many  as  A  and  one  third  as  many  as  B,  and  D  as  many 
as  B  and  C  together ;  how  many  has  each  ? 

Ans.  A  4,  B  12,  C  8,  and  D  20. 

23.  A  man  has  4  children,  the  sum  of  whose  ages  is  48  years, 
and  the  common  difierence  of  their  ages  is  equal  to  twice  that  of 
the  youngest;  required  their  ages.     Ans.  3,  9,  15,  and  21  years. 

24.  Divide  the  number  55  into  two  parts,  in  proportion  to  each 
other  as  2  to  3. 

Let  2x=  one  part ;  then  Sx^=  the  other,  since  2x  is  to  3a;  as  2 
is  to  3.  2a;+3a:=55 

5a;=55 
a;=ll 
2a;=22)   , 
3x=33  I  ^"«- 
Or  thus:  Let  x=  one  part;  then  55 — x^=  the  other. 
By  the  question,  x  :  55 — x  :  :  2  :  3.     Then,  since,  in  every  pro- 
portion, the  product  of  the  means  is  equal  to  the  product  of  the 
extremes,  we  have     3a;=2(55 — x)=110 — 2x 
5x=110 
a:=22,  and  55 — x— 33,  as  before. 

Or  thus :  Let  x=  one  part,  then  -^=^  the  other. 
And  x+  -r=55. 

2x+3a:=110,  from  which  a:=22,  and  ^=33. 

The  first  method  avoids  fractions,  and  is  of  such  frequent  appli- 
cation, that  we  may  give  this  general  direction : 


SIMPLE   EQUATIONS.  127 

When  two  or  more  unknown  qiianiities  in  any  problem,  are  to  each 
other  in  a  given  ratio,  it  is  best  to  assume  each  of  them  a  multiple  of 
some  other  unknown  quantity,  so  that  they  shall  have  to  each  other 
the  given  ratio. 

25.  The  sum  of  two  numbers  is  60,  and  the  less  is  to  the  greater 
as  5  to  7  ;  what  are  the  numbers  ?  Ans.  25  and  35. 

26.  Divide  the  number  60  into  3  parts,  which  shall  be  in  pro- 
portion to  each  other  as  2,  3,  and  5.  Ans.  12,  18,  and  30. 

27.  Divide  the  number  92  into  4  parts,  in  proportion  to  each 
other  as  the  numbers  3,  5,  7,  and  8.       Ans.  12,  20,  28,  and  32. 

28.  Divide  the  number  60  into  3  such  parts,  that  I  of  the  first, 
\  of  the  second,  and  \  of  the  third,  shall  all  be  equal  to  each  other. 

Ans.  12,  18,  and  30. 
This  question  is  most  conveniently  solved,  by  putting  2x,  3a:, 
and  5x  for  the  parts,  since  the  o,  |,  and  i  of  these  are  respec- 
tively equal  to  each  other.  > 


29.  What  number  is  that  whose  half,  third,  and  fourth  part  are 
together  equal  to  65  ?  Ans.  60. 

30.  What  number  is  that,  i  of  which  is  greater  than    |  by  4  ? 

Ans.  70. 

31.  The  age  of  B  is  two  and  four  fifth  times  the  age  of  A,  and 
tlie  sum  of  their  ages  is  76  years ;  what  is  the  age  of  each  ? 

Ans.  A  20,  B  56  years. 

32.  Divide  88  dollars  between  A,  B,  and  C,  giving  to  B  |,  and 
to  C  I  as  much  as  to  A.  Ans.  A  $42,  B  $28,  and  C$18. 

33.  Divide  440  dollars  between  three  persons,  A,  B,  and  C,  so 
that  the  share  of  A  may  be  f  that  of  B,  and  the  share  of  B  |  that 
of  C.  Ans.  A's  share  $90,  B's  $150,  and  C's  $200. 

34.  Four  towns  are  situated  in  the  order  of  the  letters  A,  B,  C, 
D.  The  distance  from  A  to  D  is  120  miles  ;  the  distance  from  A 
to  B  is  to  the  distance  from  B  to  C  as  3  to  5  ;  and  one  third  of  the 
distance  from  A  to  B,  added  to  the  distance  from  B  to  C,  is  three 
times  the  distance  from  C  to  D;  how  far  are  the  towns  apart? 

Ans.  A  to  B,  36  miles ;  B  to  C,  60  miles  ;  C  to  D,  24  miles. 

35.  A  merchant  having  engaged  in  trade  with  a  certain  capital, 
lost  \  of  it  the  1st  year ;  the  2d  year  he  gained  a  sum  equal  to  § 
of  what  remained  at  the  close  of  the  1st  year ;  the  3d  year  he  lost 
4  of  what  he  had  at  the  close  of  the  2d  year,  when  he  was  worth 
$1236.     What  was  his  original  capital  ?  Ans.  $1545. 

36.  The  rent  of  a  house  this  year,  is  greater,  by  5  per  cent., 
than  it  was  last  year ;  this  year  the  rent  is  168  dollars ;  what  was 
it  last  year?        '  Ans.  $160. 


128  RAT'S   ALGEBRA,    PART   FIRST. 

37.  Divide  the  number  32  into  2  parts,  so  that  the  greater  shall 
exceed  the  less  by  6.  Ans.  13  and  19. 

38.  At  an  election,  the  number  of  votes  given  for  two  candi- 
dates, was  256 ;  the  successful  candidate  had  a  majority  of  50 
votes  ;  how  many  votes  had  each  ?  Ans.  153  and  103. 

39.  Divide  1520  dollars  between  three  persons.  A,  B,  C,  so  that 
B  may  receive  100  dollars  more  than  A,  and  C  270  dollars  more 
than  B  ;  what  is  the  share  of  each  ? 

Ans.  A  $350,  B  $450,  and  C  $720. 

40.  A  company  of  90  persons  consists  of  men,  women,  and 
children  ;  the  men  are  4  more  than  the  women,  and  the  children 
are  10  more  than  both  men  and  Avomen ;  what  is  the  number  of 
each?  Ans.  18  women,  22  men,  and  50  children. 

41.  After  cutting  off  a  certain  quantity  of  cloth  from  a  piece 
containing  45  yards,  it  was  found  that  there  remained  9  yards 
less  than  had  been  cut  off;  how  many  yards  had  been  cut  off? 

Ans.  27. 

42.  What  number  is  that,  which,  being  multiplied  by  7,  gives 
a  product  as  much  greater  than  20,  as  the  number  itself  is. less 
than  20  ?  Ans.  5. 

43.  A  person  dying,  left  an  estate  df  G500  dollars,  to  be  divided 
between  his  widow,  2  sons,  and  3  daughters,  so  that  each  son 
shall  receive  twice  as  much  as  a  daughter,  and  the  widow  500 
dollars  less  than  all  her  children  together ;  required  the  share  of 
the  Avidow,  and  of  each  son  and  daughter. 

Ans.   Widow  $3000,  each  son  $1000,  and  each  daughter  $500. 


44.  Two  men  set  out  at  the  same  time,  one  from  London,  and 
the  other  from  Edinburgh  ;  one  goes  20,  and  the  other  30  miles  a 
day  ;  in  how  many  days  will  they  meet,  the  distance  being  400 
miles?  Ans.  8  days. 

45.  Two  persons,  A  and  B,  depart  from  the  same  place,  to  go 
in  the  same  direction ;  B  travels  at  the  rate  of  3,  and  A  at  the 
rate  of  5  miles  an  hour,  but  B  has  the  start  of  A  10  hours ;  in 
how  many  hours  will  A  overtake  B?  Ans.  15. 

46.  What  number  is  that,  of  which  one  half  and  one  third  of 
it  diminished  by  44,  is  equal  to  one  fifth  of  it  diminished  by  6  ? 

Ans.  60. 

47.  A  person  being  asked  the  time  of  day,  replied,  "If,  to  the 
time  past  noon,  there  be  added  its  -J,  -],  and  §,  the  sum  will  be 
equal  to  g  of  the  time  to  midnight ;  required  the  hour. 

Ans.  50  min.  P.  M. 


SIMPLE    EQUATIONS.  129 


48.  Divide  the  number  120  into  two  such  parts,  that  the  smaller 
may  be  contained  in  the  greater  H  times.  Ans.  48  and  72. 

49.  "I  have  a  certain  number  in  my  mind,"  said  A  to  B;  "if 
I  multiply  it  by  7,  add  3  to  the  product,  divide  this  by  2,  and  sub- 
tract 4  from  the  quotient,  the  remainder  is  15."  What  is  the 
number?  Ans.  5. 

50.  What  number  is  that,  which,  if  you  multiply  it  by  5,  sub- 
tract 24  from  the  product,  divide  the  remainder  by  6,  and  add  13 
to  the  quotient,  will  give  the  number  itself?  Ans  54. 

51.  Two  persons,  A  and  B,  engaged  in  trade,  the  capital  of  B 
being  §  that  of  A ;  B  gained,  and  A  lost,  100  dollars  ;  after  which, 
if  f  of  what  A  had  left,  be  subtracted  from  what  B  now  has,  the 
remainder  will  be  134  dollars ;  Avith  what  capital  did  each  com- 
mence ?  Ans.  A  $786,  B  $524. 

52.  A  man  having  spent  3  dollars  more  than  |  of  his  money, 
had  7  dollars  more  than  j  of  it  left ;  how  many  dollars  had  he  at 
first?  Ans.  $75. 

53.  Tavo  men,  A  and  B,  have  the  same  annual  income  ;  A  saves 
^  of  his,  but  B  spends  25  dollars  per  annum  more  than  A,  and  at 
the  end  of  5  years  finds  he  has  saved  200  dollars ;  what  is  the 
annual  income  of  each?  Ans.  $325. 

54.  In  the  composition  of  a  quantity  of  gunpowder,  f  of  the 
whole,  plus  10  pounds,  was  niter;  5^^  of  the  whole,  plus  1  pound, 
was  sulphur;  and  ^  of  the  whole,  minus  17  pounds,  was  charcoal ; 
how  many  pounds  of  gunpowder  were  there  ?  Ans.  69tb. 

55.  A  person  bought  a  chaise,  horse,  and  harness,  for  245  dol- 
lars ;  the  horse  cost  3  times  as  much  as  the  harness,  and  the  chaise 
cost  19  dollars  less  than  2f  times  as  much  as  both  horse  and  har- 
ness ;  what  was  the  cost  of  each  ? 

Ans.  Harness  $18,  horse  $54,  chaise  $173. 

56.  What  two  numbers  are  as  3  to  4,  to  each  of  which,  if  4  be 
added,  the  sums  will  be  to  each  other  as  5  to  6?        Ans.  6  and  8. 

57.  What  two  numbers  are  as  2  to  5,  from  each  of  which,  if  2 
be  subtracted,  the  remainders  will  be  to  each  other  as  3  to  8  ? 

Ans.  20  and  50. 

58.  The  ages  of  two  brothers  are  now  25  and  30  years,  so  that 
their  ages  are  as  5  to  6 ;  in  how  many  years  will  their  ages  be  as 
8  to  9?  Ans.  15. 

How  many  years  since  their  ages  were  as  1  to  2  ?       A.  20  yrs. 

59.  A  cistern  has  3  pipes  to  fill  it ;  by  the  first,  it  can  be  filled 
in  1^  hours,  by  the  second,  in  33  hours,  and  by  the  third,  in  5 
hours ;  in  what  time  can  it  be  filled,  by  all  three  running  at  once? 

Ans.  48  min. 


130  RAY'S   ALGEBRA,    PART   FIRST. 

60.  Find  the  time  in  which  A,  B,  and  C  together,  can  perform 
a  piece  of  work,  which  requires  7,  6,  and  9  days  respectively, 
when  done  singly.  Ans.  2§5  days. 

61.  From  a  certain  sum  I  took  one  third  part,  and  put  in  its 
stead  50  dollars;  next,  from  this  sum  I  took  the  tenth  part,  and 
put  in  its  stead  37  dollars ;  I  then  counted  the  money,  and  found 
I  had  100  dollars;  what  was  the  original  sum?  Ans.  $30. 

62.  A  teacher  spent  §  of  his  yearly  salary  for  board  and  lodging, 
I  of  the  remainder  for  clothes,  and  I  of  what  remained,  for  books, 
and  still  saved  120  dollars  per  annum;  what  was  his  salary? 

Ans.  $375. 

03.  A  laborer  was  engaged  for  a  year,  at  80  dollars  and  a  suit 

of  clothes ;  after  he  had  served  7  months,  he  left,  and  received  for 

his  wages,  the  clothes  and  35  dollars ;  what  was  the  value  of  the 

suit  of  clothes  ?  Ans.  $28. 

64.  A  man  and  his  wife  can  drink  a  cask  of  wine  in  6  days, 
and  the  man  alone  can  drink  it  in  10  days;  how  many  days  will 
it  last  the  woman  ?  Ans.  15. 

65.  A  steamboat,  that  can  run  15  miles  per  hour  with  the  cur- 
rent, and  10  miles  per  hour  against  it,  requires  25  hours  to  go 
from  Cincinnati  to  Louisville,  and  return ;  what  is  the  distance 
between  those  cities  ?  Ans.  150  miles. 

60.  A  and  B  engaged  in  a  speculation ;  A  with  240  dollars, 
and  B  with  96  dollars ;  A  lost  twice  as  much  as  B,  and,  upon  set- 
tling their  accounts,  it  appeared,  that  A  had  3  times  as  much 
remaining  as  B  ;  what  did  each  lose  ?      Ans.  A  $90,  and  B  $48. 

67.  In  a  mixture  of  wine  and  water,  \  the  whole,  plus  25  gal- 
lons, was  wine,  and  \  of  the  whole,  minus  5  gallons,  was  water ; 
required  the  quantity  of  each  in  the  mixture. 

Ans.  85  galls,  of  wine,  and  35  galls,  of  water. 

68.  It  is  required  to  divide  the  number  91  into  2  such  parts, 
that  the  greater,  being  divided  by  their  difference,  the  quotient 
will  be  7.  Ans.  49  and  42. 

69.  It  is  required  to  divide  the  number  72  into  4  such  parts, 
that  if  the  first  be  increased  by  2,  the  second  diminished  by  2,  the 
third  multiplied  by  2,  and  the  fourth  divided  by  2,  the  sum,  the 
difference,  the  product,  and  the  quotient  shall  all  be  equal. 

Ans.  14,  18,  8,  and  32. 
Let  the  four  parts  be  represented  by  x — 2,  x-\-2,  l^,  and  2jc. 

70.  A  merchant  having  cut  19  yards  from  each  of  3  equal  pieces 
of  silk,  and  17  from  another  of  the  same  length,  found,  that  the 
remnants  taken  together,  measured  142  yards ;  what  was  the 
length  of  each  piece?  -  Ans.  54  yds. 


SIMPLE   EQUATIONS.  131 


71.  Suppose,  that  for  every  10  sheep  a  farmer  keeps,  he  should 
plow  an  acre  of  land,  and  allow  1  acre  of  pasture  for  every  4 
sheep;  how  many  sheep  can  the  person  keep,  who  farms  161 
acres  ?  Ans.  460. 

72.  It  is  required  to  divide  the  number  34  into  2  such  parts, 
that  if  18  be  subtracted  from  the  greater,  and  the  less  be  sub- 
tracted from  18,  the  first  remainder  shall  be  to  the  second  as  2 
to  3.  Ans.  22  and  12. 

73.  A  person  was  desirous  of  giving  3  cents  a  piece  to  some 
beggars,  but  found  that  he  had  not  money  enough  in  his  pocket 
by  8  cents ;  he  therefore  gave  each  of  them  2  cents,  and  then  had 
3  cents  remaining  ;  required  the  number  of  beggars.        Ans.  1 1 . 

74.  A  father  distributed  a  number  of  apples  among  his  chil- 
dren, as  follows :  to  the  first  he  gave  ^  the  whole  number,  less  8  ; 
to  the  second  ^  the  remainder,  diminished  by  8 ;  and  in  the  same 
manner,  with  the  third  and  fourth  ;  after  which,  he  had  20  apples 
remaining  for  the  fifth  ;  how  many  apples  did  he  distribute  ? 

Ans.  80. 

75.  A  could  reap  a  field  in  20  days,  but  if  B  assisted  him  for  6 
days,  he  could  reap  it  in  16  days;  in  how  many  days  could  B 
reap  it  alone  ?  Ans.  30  days. 

76.  There  are  two  numbers  in  the  proportion  of  ^  to  f,  which, 
being  increased  respectively,  by  6  and  5,  are  in  the  proportion  of 
§  to  2  ;  required  the  numbers.  Ans.  30  and  40. 

77.  When  the  price  of  a  bushel  of  barley  wanted  but  3  cents 
to  be  to  the  price  of  a  bushel  of  oats  as  8  to  5,  nine  bushels  of 
oats  were  received  as  an  equivalent  for  4  bushels  of  barley  and 
90  cents  in  money  ;  what  was  the  price  of  a  bushel  of  each  ? 

Ans.  Oats  30  cts.,  and  barley  45  cts. 

78.  Four  places  are  situated  in  the  order  of  the  4  letters.  A,  B, 
C,  and  D  ;  the  distance  fro!n  A  to  D  is  34  miles ;  the  distance 
from  A  to  B  is  to  the  distance  from  C  to  D,  as  2  to  3  ;  and  |  the 
distance  from  A  to  B,  added  to  ^-  the  distance  from  C  to  D,  is  3 
times  the  distance  from  B  to  C.     Required  the  respective  distances. 

Ans.  A  to  B  12,  B  to  C  4,  and  C  to  D  18  miles. 

79.  The  ingredients  of  a  loaf  of  bread  are  rice,  flour,  and  water, 
and  the  weight  of  the  whole  is  15  pounds ;  the  weight  of  the  rice 
increased  by  5  pounds,  is  f  the  weight  of  the  flour;  and  the 
weight  of  the  water  is  ^  the  weight  of  the  flour  and  rice  together; 
what  is  the  weight  of  each? 

Ans.  Rice  21b,  flour  lO^tb,  and  water  2nlb. 


132  RAY'S   ALGEBRA,    PART  FIRST. 


SIMPLE  EQUATIOIVS  C01\TAIIVIi\G  TWO  UIVKI^OWIV  QUAl^TITIES. 

Art.  157.— In  order  to  find  the  value  of  any  unknown  quantity, 
it  is  evident,  that  we  must  obtain  a  single  equation  containing  it, 
and  known  terms.  Hence,  when  we  have  two  or  more  equations, 
containing  two  or  more  unknown  quantities,  we  must  obtain  from 
them  a  single  equation  containing  only  one  unknown  quantity. 
The  method  of  doing  this,  is  termed  elimination,  w^hich  may  bo 
briefly  defined  thus:  Elimination  is  the  process  of  deducing  from 
two  or  more  equations,  containing  two  or  more  unknown  quantities, 
a  less  number  of  equations  containing  one  less  unknown  quantity. 

There  are  three  methods  of  elimination. 

1st.  Elimination  by  substitution. 

2d.    Elimination  by  comparison. 

3d.    Elimination  by  addition  and  subtraction, 

ELIMIIVATIOJV    BY    SUBSTITUTION. 

Art.  15§. — Elimination  by  substitution,  consists  in  finding  the 
value  of  one  of  the  unknown  quantities  in  o»e  of  the  equations,  in 
terms  of  the  other  unknown  quantity  and  known  terms,  and  sub- 
stituting this,  instead  of  the  quantity,  in  the  other  equation. 

To  explain  this,  suppose  we  have  the  following  equati(ms,  in 
w^hich  it  is  required  to  find  the  value  of  x  and  y. 

Note.  —  The  figures  in  the  parentheses,  are  intended  to  number  the 
equations  for  reference. 

a:+2y=17     (1.) 
2x+3?/=28     (2.) 
By  transposing  2y  in  the  equation  ( 1 ),  we  have x—\l — 2?/.     Su]> 
stituting  this  value  of  x,  instead  of  x  in  equation  (2),  we  have 
2(17-2^)+32/=28 
or,     34-4?/+3y^28 
or,     -?/=28-34 

and      x=17-2y=17-12=5. 
Hence,  when  we  have  two  equations,  containing  two  unknown 
quantities,  we  have  the  following 

RULE, 

FOR    ELIMINATION    BY    SUBSTITUTION. 

Find  an  expression  for  the  value  of  one  of  the  unknown  quan- 
tities in  either  equation,  and  substitute  this  value  in  place  of  the  same 
unknown  quantiit/  in  the  other  equation ;  there  will  thus  be  formed  a 
new  equation,  containing  only  one  unknown  quantity. 


SIMPLE    EQUATIONS. 


133 


Note. —  In  fiiuUng  an  expression  for  the  value  of  one  of  the  unknown 
quantities,  let  that  be  taken  which  is  the  least  involved. 

Find  the  values  of  the  unknown  quantities  in  each  of  the  fol- 
lowing equations. 


Ans. 


Ans. 


a:=3. 

a;  ^=5. 
2/=3. 
Ans.  a;=4. 

Ans.  a:=8. 

Ans.  a;=16. 
y=12. 


6. 


X— y=10. 

Ans.  x=25. 

5     3^- 

y=15. 

11-^=1 
5     4 

Ans.  a:=.20. 

bx-3i/=l0. 
2x     Si/ 

y~8-^- 

y=30. 
Ans.  x=21. 

3+^-26. 

y==l6. 

1.  x+5?/-.38. 
3a:4-4y=37. 

2.  2x+4y=22. 
5x-f7y=46. 

3.  3x+5//=:57. 
5a:+3y=47. 

4.  4x-3i/=2Q. 
3x—4y=lQ. 

5.  2x— 3y=~4. 

.-1=1.. 

ELIMINATION    BY     COMPARISON. 

Art.  159.— Elimination  by  comparison,  consists  in  finding  the 
value  of  the  same  unknown  quantity  in  two  different  equations, 
and  then  placing  these  values  equal  to  each  other. 

To  illustrate  this  method,  we  will  take  the  same  equations  which 
were  used  to  explain  elimination  by  substitution. 
a:+2y=17     (1.) 
2x+3^=28    (2.) 
By  transposing  2i/  in  equation  (1),  we  have  x=17^2y. 
By  transposing  3y  in  equation  (2),  and  dividing  by  2,  we  have 
_28-3// 

Placing  these  values  of  x  equal  to  each  other, 

or,  28-3y=34-4y 
or,  y=6. 
The  value  of  x  may  be  found  in  a  similar  manner,  by  first  find- 
ing the  values  of  y,  and  placing  them  equal  to  each  other.  But, 
after  having  found  the  value  of  one  of  the  unknown  quantities, 
the  value  of  the  other  may  be  found  most  readily  by  substitution, 
as  in  the  preceding  article.     Thus,  x=\l—2y=\l — 12—5. 

Review. — 157.  What  is  necessary  in  order  to  find  the  value  of  any 
unknown  quantity  ?  When  we  have  two  equations,  containing  two  unknown 
quantities,  what  is  necessary,  in  order  to  find  the  value  of  one  of  them? 
What  is  elimination  ?  How  many  methods  of  elimination  are  there  ?  158. 
In  what  does  elimination  by  substitution  consist?  What  is  the  rule  for 
elimination  by  substitution  ?  159.  In  what  does  elimination  by  compari- 
son consist? 


134 


RAY'S   ALGEBRA,    PART    FIRST. 


Hence,  when  we  have  two  equations,  containing  two  unknown 
quantities,  we  have  the  following 

RULE, 

FOR   ELIMINATION    BY    COMPARISON. 

Find  an  expression  for  the  value  of  the  same  unknown  quantity  in 
each  of  the  given  equations,  and  place  these  values  equal  to  each 
other;  there  will  thus  be  formed  a  new  equation,  containing  only  one 
unknown  quantity. 

Find  the  value  of  each  of  the  unknown  quantities  in  the  follow- 
ing equations,  by  the  preceding  rule. 


1.  .^+3^=16. 

2.  3x~\-5y=29. 

3.  5x-2y=4. 

^'2     3 

X  , 

^'  9     8-^- 


6^4 
Sx 


12. 


Ans.  x=7. 

y=S. 
Ans.  a;=8. 

y=l. 
Ans.  x=2. 

Ans.  a;=6. 

y=S. 

Ans.  x=36. 

y=24. 


6. 


4     4""^- 


3^2 


:8. 

d4. 


Ans.  a;=12. 

y=8- 

Ans.  x=45. 


9~5-^' 


2x 

7' 


Sy 
5 


y=W, 
27.       Ans.  a:=21. 


9      7 


r 


^35. 


9.^+2y-x+|^= 


10. 


3a;— 5.y_2a;+4 
2     ~    7     " 
x—2y_x+y 
4    ~  3 


6- 


42 Ans.  x=20. 

y=12. 

-1.  .   . Ans.  x=12. 

y=6. 


ELIMIIVATION   BY  ADDITION    AIVD    SUBTRACTIOIV. 

Art.  160. — Elimination  by  addition  and  subtraction,  consists 
in  multiplying  or  dividing  two  equations,  so  as  to  render  the  coef- 
ficient of  one  of  the  unknown  quantities,  the  same  in  both ;  and 
then,  by  adding  or  subtracting,  to  cause  the  term  containing  it  to 
disappear. 

To  explain  this  method,  we  will  take  the  same  equations  used 
to  illustrate  elimination  by  substitution  and  comparison, 
x+2y=l7     (1.) 
2a;+3y=28     (2.) 


SIMPLE   EQUATIONS.  135 

If  we  multiply  equation  (1)  by  2,  so  as  to  make  the  coeflficient 
of  X  the  same  as  in  the  second  equation,  we  have 

2a:+4y=34     (3.) 

2a;-f3y=28,  equation  (2)  brought  down. 

Since  the  coefficient  of  x  has  the  same  sign  in  these  equations, 
if  we  subtract,  the  terms  containing  x  will  cancel  each  other,  and 
the  resulting  equation  will  contain  only  y,  the  value  of  which  may 
then  readily  be  found.  After  this,  by  substituting  the  value  of  y, 
as  before,  the  value  of  x  is  easily  obtained. 

To  illustrate  the  method  of  eliminating,  when  the  coefficients  of 
the  unknown  quantity  to  be  eliminated,  have  contrary  signs  in  the 
two  equations,  suppose  we  have  the  following,  in  Avhich  it  is 
required  to  eliminate  y. 

3x-5y=6      (1.) 
4x+3y=37    (2.) 
It  is  obvious,  that  if  we  multiply  equation  (1)  by  3  and  (2)  by 
5,  that  the  coefficients  of  y  will  be  the  same.     Thus, 
9x-15?/=  18 
20a;+15y^l85 
adding,  29a;      =      203 
X      =      1. 
Substituting  this  value  of  x  in  equation  (2),  we  have 

28+3?/=37 

3y=  9 

y=  3 

From  this  we  see,  that  after  making  the  coefficients  of  the  quan- 
tity to  be  eliminated,  the  same  in  both  equations,  if  the  signs  are 
alike,  we  must  subtract;  but  if  they  are  unlike,  we  must  ac?c?  them. 

Hence,  when  we  have  two  equations,  containing  two  unknown 
quantities,  we  have  the  following 

RULE, 

FOR   ELIMINATION    BY    ADDITION    AND    SUBTRACTION. 

Multiply,  or  divide  the  equations,  if  necessary,  so  that  one  of  the 
unknown  quantities  will  have  the  same  coefficient  in  both.  Then  take 
the  difference,  or  the  sum  of  the  equations,  according  as  the  signs  of 
the  equal  terms  are  alike  or  unlike,  and  the  residting  equation  will 
contain  only  one  unknown  quantity. 

Remark  . — When  the  coeCficients  of  the  unknown  quantities  to  be  elim- 
inated are  prime  to  each  other,  they  may  be  equalized,  by  multiplying  each 

R  E  V I E  w. — 159.  What  is  the  rule  for  elimination  by  comparison  ?  160.  In 
what  does  elimination  by  addition  and  subtraction  consist  ?  What  is  the 
rule  for  elimination  by  addition  and  subtraction  ? 


136 


RAY'S  ALGEBRA,    PART    FIRST. 


equation  by  the  coefficient  of  the  unknown  quantity  in  the  other.  When 
the  coSfficients  are  not  prime,  find  their  least  common  multiple,  and  multiply 
each  equation  by  the  quotient  obtained  by  dividing  the  least  common  mul- 
tiple by  the  coefficient  of  the  unknown  quantity  to  be  eliminated  in  the 
other  equation. 

If  the  equations  have  fractional  coefficients,   they  ought  to  be  cleared 
before  applying  the  rule. 

Find  the  value  of  the  unknown  quantities  in  each  of  the  follow- 


ing equations,  by  the  preceding  rule. 


1.  3a:+2?/=21. 

Ans.  ic^^S. 

5.  1+1=8. 

Ans.  x=2Q 

X— 2^=— 1. 

2/=3. 

4    5 

2.  3x-2y=7. 
5ij-2x=l0. 

Ans.  x=5. 

y=15. 

3.  2x-y^3. 

Ans.  x=4. 

X       V     ^ 

3x+2i/=22. 

y=5. 

«•  3-r-^- 

Ans.  a;=12 

4.  3x+2y=19 

2x-3.v-=4. 

Ans.  x=5. 

y=2. 

6+9"^' 

y=9. 

1  '             f                          c\ 

=3 

.    .    Ans.  x=^A. 

5           2 

=0 

.    .         v=G 

2     '    10 

'    '    '    '     y     ^ 

QUESTIOIVS   PRODUCING   EQUATIOIVS    COATAINING   TWO 
UIVKIVOWJV    QUANTITIES. 

Art.  161. — The  questions  contained  in  Art.  156,  were  all  capa- 
ble of  being  solved  by  using  one  unknown  quantity;  although, 
several  of  the  examples  contained  two,  and  in  some  cases  more, 
unknown  quantities.  In  those  questions,  however,  there  was  such 
a  connection  existing  between  the  several  quantities,  that  it  was 
easy  to  express  each  one  in  terms  of  the  other.  But  it  frequently 
happens,  that  in  a  problem  containing  tAA'o  unknown  quantities, 
there  may  be  no  direct  relation  existing  betAveen  them,  by  means 
of  Avhich  either  of  them  may  be  found  in  terms  of  the  other.  In 
such  a  case,  it  becomes  necessary  to  use  a  separate  symbol  for 
each  unknown  quantity,  and  then  to  find  the  equations  containing 
these  symbols,  on  the  same  principle  as  where  there  was  but  one 
unknown  quantity ;  that  is,  in  brief,  regard  the  symbols  as  the  an- 
swer to  the  question,  and  then  proceed  in  the  same  manner  as  it  would 
be  necessary  to  do,  to  prove  the  answer.  After  the  equations  are 
obtained,  the  values  of  the  unknown  quantities  may  be  found,  by 
either  of  the  three  difierent  modes  of  elimination. 

We  shall  first  give  two  examples,  which  can  be  solved  by  using 
either  one  or  two  unknown  quantities. 


SIMPLE    EQUATIONS.  137 

In  general,  no  more  symbols  should  be  used,  than  are  really 
necessary ;  unless,  by  using  them,  the  solution  is  rendered  more 
simple. 

1.  Given,  the  sum  of  two  numbers  equal  to  25,  and  their  differ- 
ence equal  to  9,  to  find  the  numbers. 

Solution,  by  using  one  unknown  quantity. 
Let  a:=  the  less  number  ;  then  a;+9=  the  greater. 
And  a,'+a:+9=25. 
2x=16 
a;=8,  the  less  number;  and  a:+9=17,  the  greater. 
Solution,  by  using  two  unknown  quantities. 
Let  a;=  the  greater,  and  y--^  the  less. 
Theiix+i/^2D     (1.) 
And   X  -y=  9     (2.) 

2a:=34,  by  adding  the  two  equations  together. 

x=\l,  the  greater  number. 
2^=16,  by  subtracting  equation  (2)  from  equation  (1). 
y=i   8,  the  less  number. 

2.  The  sum  of  two  numbers  is  44,  and  they  are  to  each  othtr 
as  5  to  6  ;  required  the  numbers. 

Solution,  by  using  one  unknown  quantity. 
Lot  5x=  the  less  number ;  then  Qx=  the  greater. 
And  5a;+()a:=44. 
1  la;=44 
a;=4 

5x=20,  the  less  number. 
6x=24,  the  greater  number. 
Solution,  by  using  two  unknown  quantities. 
Let  a:=  the  less  number,  and  ?/=  the  greater. 
Thenx+?/=:44     (1.) 
And  X  :  y  :  :  5  :  6 
or,  6x=^5//     (2.)     by  multiplying  means  and  extremes. 
6x+6?/=264     (3.)     by  multiplying  equation  (1)  by  6. 
6?/=264 — by,  by  subtracting  equation  (2)  from  (3). 
ll2/=264 

y=24  and  x^44— ?/=20. 
Several  of  the  following  questions  may  also  be  solved  by  using 
only  one  unknown  quantity. 

3.  There  is  a  certain  number  consisting  of  two  places  of  figures  ; 
the  sum  of  the  figures  is  equal  to  6,  and,  if  from  the  double  of  the 

R  E  V  I  E  AV. — 161.  In  solving  questions,  when  docs  it  become  necessary  to 
use  a  separate  symbol  for  each  unknown  quantity  ?  How  are  the  equations 
formed,  from  which  the  values  of  the  unknown  quantities  are  to  be  obtained  ? 

12 


138  KAY'S  ALGEBKA,  PAKT  FIKST. 


number  6  be  subtracted,  the  remainder  is  a  number  whope  digits  aje 
those  of  the  former  in  an  inverted  order;  required  the  number? 

In  solving  questions  of  this  kind,  the  pupil  must  be  reminded, 
that  any  number  consisting  of  two  placer,  of  figures,  is  equal  to  10 
times  the  figure  in  the  ten's  place,  plus  the  iigure  in  the  unit's 
place.  Thus,  23  is  equal  to  10X2-[  3.  In  a  similar  manner,  325 
is  equal  to  100X3+10X24  5. 

Let  x=  the  digit  in  the  place  of  tens,  and  y=  that  in  tlie  place 
of  units. 

Then  lOx-f  2/==  tlie  number. 

And    lOy-\-x=  the  number,  with  the  digits  inverted.       , 

Then  x+y^6     (1.) 

And2(10.i-+y)— 6=102/+x     (2.) 
or,   20.r+2i/— 6=10?/+a:. 

19x--8y+6 

8x'=^ — 82/+48,  from  equation   (1),  by  multiplying  by  8 
and  transposing. 

27x=54  by  adding. 
x=2 

2/r=6— 2=4.    Ans.  24. 
4.  What  two  numbers  are  those,  to  which  if  5  be  added,  the 
sums  will  be  to  each  other  as  5  to  6 ;  but,  if  5  be  subtracted  from 
each,  the  remainders  will  be  to  each  other  as  3  to  4? 

By  the  conditions  of  the  question,  we  have  the  following  pro- 
portions :  ?;+5  : 2/+5  : :  5  :  6 
a:— 5  :  2/— 5  : :  3  :  4. 
Since,  in  every  proportion,  the  product  of  the  means  is  equal  to 
the  product  of  the  extremes,  we  have  the  two  equations 

6(x-f5)=5(2/+5) 
4(x-5)=3(2/-5) 
From  these  equations,  the  values  of  x  and  y  are  readily  found  to 
be  20  and  25. 

Remark.— Instead  of  saying,  that  the  two  sums  will  be  to  each  other 
as  5  to  6,  it  will  be  the  same  to  say,  that  the  quotient  of  the  second  divided 
by  the  first,  is  equal  to  ^,  since  six  divided  by  5,  expresses  the  ratio  of  5  to  6. 
This  would  give  the  following  equations  : 

a;+5     5  x— 5     3 

which  may  be  readily  obtained  from  those  given  above. 

Note.— In  solving  the  following  q\iestions,  after  finding  the  equations, 
the  values  of  the  unknown  quantities  may  be  found  by  either  of  the  thi-ee 
methods  of  elimination. 


SIMPLE   EQUATIONS.  139 

5.  A  grocer  sold  to  one  person  5  pounds  of  coffee  and  3  pounds 
of  sugar,  for  79  cents ;  and  to  another,  at  the  same  prices,  3 
pounds  of  coffee  and  5  pounds  of  sugar,  for  73  cents ;  M'hat  was 
the  price  of  a  pound  of  each?        Ans.  Coffee  11  cts.,  sugar  8  cts. 

G.  A  farmer  sold  to  one  person  9  horses  and  7  cows,  for  300 
dollars  ;  and  to  another,  at  the  same  prices,  6  horses  and  13  cows, 
for  the  same  sum  ;  what  was  the  price  of  each  ? 

Ans.  Horses  $24,  and  cows  $12  each. 

7.  A  vintner  sold  at  one  time,  20  dozen  of  port  wine  and  30  of 
fherry,  and  for  the  whole  received  120  dollars;  and,  at  another, 
30  dozen  of  port  and  25  of  sherry,  at  the  same  prices  as  before, 
f)r  140  dollars;  what  was  the  price  of  a  dozen  of  each  sort  of 
wine  ?  Ans.  Port  $S,  and  sherry  $2  per  doz. 

8.  It  is  required  to  find  two  numbers,  such  that  h  of  the  first 
and  I  of  the  second  shall  be  22,  and  |  of  the  first  and  -^  of  the 
second  shall  be  12.  Ans.  24  and  30. 

9.  If  the  greater  of  two  numbers  be  added  to  3  of  the  less,  the  sum 
will  be  37;  but  if  the  less  be  diminished  by  |  of  the  greater,  the 
difference  will  be  20  ;  what  are  the  numbers  ?       Ans.  28  and  27. 

10.  What  two  numbers  are  those,  such  that  A  of  the  first  dimin- 
ished by  I  of  the  second,  shall  be  5,  and  |  of  the  first  diminished 
by  -5  of  the  second,  shall  be  2?  Ans.  20  and  15. 

11.  A  farmer  has  2  horses,  and  a  saddle  worth  25  dollars  ;  noAV, 
if  the  saddle  be  put  on  the  first  horse,  his  value  will  be  double 
that  of  the  second ;  but,  if  the  saddle  be  put  on  the  second  horse, 
]iis  value  will  be  three  times  that  of  the  first.  Required  the  value 
of  each  horse.  Ans.  First  $15,  second  $20. 

12.  A  and  B  are  in  trade  together  with  different  sums  ;  if  50 
dollars  be  added  to  A's  property,  and  20  dollars  taken  from  B's, 
they  will  have  the  same  sum ;  and  if  A's  property  was  3  times, 
and  B's  5  times  as  great  as  each  really  is,  they  would  together 
have  2350  dollars;  how  much  has  each?  Ans.  A  $250,  B  $320. 

13.  A  has  two  vessels  containing  wine,  and  finds,  that  |  of  the 
first  contains  96  gallons  less  than  |  of  the  second ;  and  that  g  of 
the  second  contains  as  much  as  y  of  the  first ;  liow  much  does  each 
vessel  hold?  Ans.  720  and  512  galls. 

14.  There  is  a  number  consisting  of  two  digits,  Avhich,  divided 
by  their  sum,  gives  a  quotient,  7 ;  but  if  the  digits  be  written  in 
an  inverse  order,  and  the  number  so  arising,  be  divided  by  their 
6um  increased  by  4,  the  quotient  Avill  be  3.     Required  the  number. 

Ans.  84. 

15.  If  we  add  8  to  the  numerator  of  a  certain  fraction,  its  value 
becomes  2 ;  and  if  we  subtract  5  from  the  denominator,  its  value 
becomes  3  ;  required  the  fraction.  Ans.  ^- 


140  RAY'S   ALGEBRA,    PART   FIRST. 


16.  If  to  the  ages  of  A  and  B  18  be  added,  the  result  will  be 
double  the  age  of  A  ;  but,  if  from  their  difference  6  be  subtracted, 
the  result  will  be  the  age  of  B ;  required  their  ages. 

Ans.  A  30,  B  12  yrs. 

17.  There  are  two  numbers  whose  sum  is  37,  and  if  3  times 
the  less  be  subtracted  from  4  times  the  greater,  and  the  difference 
divided  by  6,  the  quotient  will  be  6 ;  what  are  the  numbers  ? 

Ans.  16  and  21. 

18.  It  is  required  to  find  a  fraction,  such  that  if  3  be  subtracted 
from  the  numerator  and  denominator,  the  value  will  be  ^ ;  and  if  5 
be  added  to  the  numerator  and  denominator,  the  value  will  be  |. 

Ans.  -j-^. 

19.  A  father  gave  his  two  sons,  A  and  B,  together  2400  dollars, 
to  engage  in  trade ;  at  the  close  of  the  year,  A  has  lost  |  of  his 
capital,  while  B,  having  gained  a  sum  equal  to  {  of  his  capital, 
finds  that  his  money  is  just  equal  to  that  of  his  brother;  what  was 
the  sum  given  by  the  father  to  each  ?       Ans.  A  $1500,  B  $900. 

20.  If  from  the  greater  of  two  numbers  1  be  subtracted,  the 
remainder  will  be  equal  to  4  times  the  less ;  but,  if  to  the  less  3 
be  added,  the  sum  will  be  -3  of  the  greater  ;  required  the  numbers. 

Ans.  8  and  33. 

21.  A  said  to  B,  "  Give  me  100  dollars,  and  then  I  shall  have 
as  much  as  you."  B  said  to  A,  "  Give  me  100  dollars,  and  then  I 
shall  have  twice  as  much  as  you."     How  many  dollars  had  each? 

Ans.  A  $500,  B  $700. 

22.  If  the  greater  of  two  numbers  be  multiplied  by  5,  and  the 
less  by  7,  the  sum  of  their  products  is  198 ;  but  if  the  greater  be 
divided  by  5,  and  the  less  by  7,  the  sum  of  their  quotients  is  6; 
what  are  the  numbers?  Ans.  20  and  14. 

23  Seven  years  ago  the  age  of  A  was  just  three  times  that  of 
B  ;  and  seven  years  hence,  A's  age  will  be  just  double  that  of  B; 
what  are  their  ages?  Ans.  A's  49,  B's  21  yrs. 

24.  There  is  a  certain  number  consisting  of  two  places  of  figures, 
which  being  divided  by  the  sum  of  its  digits,  the  quotient  is  4, 
and  if  27  be  added  to  it,  the  digits  will  be  inverted ;  required  the 
number.  Ans.  36. 

25.  A  grocer  has  two  kinds  of  sugar,  of  such  quality  that  one 
pound  of  each  are  together  worth  20  cents;  but  if  3  pounds  of 
the  first,  and  5  pounds  of  the  second  kind  be  mixed,  a  pound  of 
the  mixture  will  be  worth  1 1  cents ;  what  is  the  value  of  a  pound 
of  each  sort?  Ans.  6  cts.,  and  14  cts. 

26.  A  boy  lays  out  84  cents  for  lemons  and  oranges,  giving  3 
cents  a  piece  for  the  lemons,  and  5  cents  a  piece  for  the  oranges ; 
he  afterwai'd  sold  \  of  the  lemons  and  -3  of  the  oranges,  for  40 


SIMPLE   EQUATIONS.  141 

cents,  and  by  so  doing  cleared  8  cents  on  what  he  sold  ;  what 
number  of  each  did  he  purchase  ? 

Ans.  8  lemons  and  12  oranges. 

27.  A  person  spends  30  cents  for  peaches  and  apples,  buying 
his  peaches  at  4,  and  his  apples  at  5  for  a  cent;  he  afterward 
sells  -A  of  his  peaches,  and  |  of  his  apples,  at  the  same  rate  he 
bought  them,  for  13  cents  ;  how  many  of  each  did  he  buy? 

Ans.  72  peaches  and  60  apples. 

28.  A  OAves  500  dollars,  and  B  owes  690  doU.ars,  but  neither  has 
sufficient  money  to  pay  his  debts.  A  said  to  B,  "  Lend  me  i  of 
your  money,  and  1  shall  have  enough  to  discharge  my  debts.'' 
B  said  to  A,  "  Lend  me  \  of  your  money,  and  I  can  pay  mine." 
How  much  money  has  each?  Ans.  A  .$400,  B  $500. 

29.  A  merchant  bought  two  pieces  of  cloth  for  236  dollars,  the 
first  piece  at  4,  and  the  second  at  7  dollars  per  yard  ;  but  the  cloth 
getting  damaged,  he  sold  -|  of  the  first  piece,  and  -|  of  the  second, 
for  160  dollars,  by  which  he  lost  8  dollars  on  what  he  sold;  what 
Avas  the  number  of  j-ards  in  each  piece? 

Ans.  24  yards  in  the  first,  and  20  yards  in  the  second. 

30.  A  son  said  to  his  father,  "  Hoav  old  are  we  ?  "  The  fiither 
replied,  "  Six  years  ago  my  age  Avas  3-3-  times  yours,  but  3  years 
hence,  my  age  will  be  only  21-  times  yours."  Required  the  age 
of  each.  Ans.  Father's  age  36,  son's  15  yrs. 

31.  A  person  has  two  horses,  and  two  saddles,  one  of  Avhich  cost 
50,  and  the  other  2  dollars.  If  he  places  the  best  saddle  upon  the 
first  horse,  and  the  other  on  the  second,  then  the  latter  is  Avorth  8 
dollars  less  than  the  former ;  but  if  he  puts  the  worst  saddle  upon 
tlie  first,  and  the  best  upon  the  second  horse,  then  the  A'alue  of  the 
latter  is  to  that  of  the  former  as  15  to  4.  Required  the  A^alue  of 
each  horse.  Ans.  First  .$30,  second  $70. 

32.  A  fixrmer  haA'ing  mixed  a  certain  number  of  bushels  of  oats 
and  rye,  fi)und,  that  if  he  had  mixed  6  bushels  more  of  each,  he 
Avould  ha\'e  mixed  7  bushels  of  oats  for  CA'cry  6  of  rye ;  but  if  he 
liad  mixed  6  bushels  less  of  each,  he  Avould  have  put  in  6  bushels 
(;f  oats  for  every  5  of  rye.  How  many  bushels  of  each  did  ho 
mix?  Ans.  Oats  78,  rye  iiQ  bu. 

33.  A  person  haA'ing  laid  out  a  rectangular  yard,  observed,  that 
if  each  side  had  been  4  yards  longer,  the  length  would  have  been 
to  the  breadth,  as  5  to  4 ;  but,  if  each  had  been  4  yards  shorter, 
the  length  Avould  have  been  to  the  breadth,  as  4  to  3  ;  required  the 
length  of  the  sides.  Ans.  Length,  36,  breadth  28  yards. 

34.  A  farmer  rents  a  farm  for  245  dollars  per  annum  ;  the  tilla- 
ble land  being  valued  at  2  dollars  an  acre,  and  the  pasture  at  1 
d  >l]ar  and  40  cents  an  acre :  now  the  number  of  acres  tillable,  is 


142  RAY'S   ALGEBRA,    PART   FIRST. 

to  the  excess  of  the  tillable  above  the  pasture,  as  14  to  9  ;  how 
many  were  there  of  each  ?  A.  Tillable  98,  pasture  35  acres. 

35.  Two  shepherds,  A  and  B,  are  intrusted  with  the  charge  of 
two  flocks  of  sheep ;  at  the  end  of  the  first  year,  it  is  found,  that 
A's  flock  has  increased  10,  and  B's  diminished  20,  when  their 
numbers  are  to  each  other,  as  4  to  3  ;  during  the  second  year,  A's 
flock  loses  20,  and  B's  gains  10,  when  their  numbers  are  to  each 
other  as  6  to  7.     Required  the  number  in  each  flock  at  first. 

Ans.  A's  had  70,  and  B's  80  sheep. 

36.  After  drawing  15  gallons  from  each  of  2  casks  of  wine,  the 
quantity  remaining  in  the  first,  is  f  of  that  in  the  second ;  after 
drawing  25  gallons  more  from  each,  the  quantity  left  in  the  first, 
is  only  half  that  in  the  second.  Required  the  number  of  gallons 
in  each  before  the  first  drawing.  Ans.  65  and  90  galls. 

37.  There  is  a  fraction,  such  that  if  1  be  added  to  the  numera- 
tor, and  the  numerator  to  the  denominator,  its  value  will  be  \  ; 
but  if  the  denominator  be  increased  by  unity,  and  the  numerator 
by  the  denominator,  its  value  will  be  f  ;  find  it.  Ans.  fj- 

38.  Find  two  numbers  in  the  ratio  of  5  to  7,  to  which  two  other 
required  numbers,  in  the  ratio  of  3  to  5,  being  respectively  added, 
the  sums  shall  be  in  the  ratio  of  9  to  13,  and  the  difference  of  their 
sums  equal  to  16.  Ans.  30  and  42,  6  and  10. 

Let  the  first  two  numbers  be  represented  by  5x  and  7x,  and  the 
other  two  by  Sij  and  5y. 

39.  A  farmer,  with  28  bushels  of  barley,  worth  28  cents  per 
bushel,  would  mix  rye  at  36  cents,  and  wheat  at  48  cents  per 
bushel,  so  that  the  whole  mixture  may  consist  of  100  bushels,  and 
be  worth  40  cents  a  bushel ;  how  many  bushels  of  rye,  and  how 
many  of  wheat  must  be  mixed  with  the  barley  ? 

Ans.  Rye  20,  and  wheat  52  bu. 

40.  Two  loaded  wagons  were  weighed,  and  their  Aveights  were 
found  to  be  in  the  ratio  of  4  to  5  ;  part  of  their  loads,  which  were 
in  the  ratio  of  6  to  7,  being  taken  out,  their  weights  were  then 
found  to  be  in  the  ratio  of  2  to  3,  and  the  sum  of  their  weights 
was  then  1 0  tons  ;  what  were  their  weights  at  first  ? 

Ans.  16  and  20  tons. 

41.  A  person  had  two  casks  and  a  certain  quantity  of  wine  in 
each ;  in  order  to  have  the  same  quantity  in  each  cask,  he  poured 
as  much  out  of  the  first  cask  into  the  second  as  it  already  con- 
tained ;  he  next  poured  as  much  out  of  the  second,  into  the  first, 
as  it  then  contained ;  and  lastly,  he  poured  out  as  much  from  the 
first  into  the  second,  as  there  was  remaining  in  it ;  after  this,  ho 
had  16  gallons  in  each  cask;  how  many  gallons  did  each  contain 
at  first?  ^      Ans.  First  22,  and  second  10  galls. 


SIMPLE    EQUATIONS.  143 


SIMPLE  EQUATIOIVS,  COIVTAIIVIXG  THREE  OR  MORE    LAKXOWJV 
QLAIVTITIES. 

Art.  162.— Equations  involving  three  or  more  unknown  quan- 
tities may  be  solved,  by  either  of  the  three  methods  of  elimination 
explained  in  the  preceding  Article,  as  we  shall  now  proceed  to 
show,  by  solving  an  example  by  each  of  these  methods. 

Suppose  we  have  the  three  following  equations,  in  which  it  ia 
required  to  find  the  values  of  x,  y,  and  z. 

x^2y+  2=20     (1.) 
2x+  ?/+32=31     (2.) 
3x+4?/+2z=44    (3.) 
Solution  by  substitution. 
From  equation  (1),  x=20—2i/—z. 
Substituting  this  in  equation  (2),  we  have 

2(20-2y-2)+y+32=31. 
or,  40-4?/— 22+?/+32---31. 
Sy-z=9  _  (4.) 
Substituting  the  same  value  of  x  in  equation  (3),  we  have 
3{20-2i/—z)-^4y+2z=44. 
or,  Q0-Qy-Sz+4y+2z=44. 
2ij-^z=l6     (5.) 
Sy-z=9       (4.) 
Here  the  values  of  y  and  z  are  readily  found  by  the  rule.  Art. 
158,  to  be  5  and  6  ;  then  substituting  these  values  in  equation  (1), 
we  find  x=4. 

Solution  by  comparison. 

From  equation  (1),  ar=20 — 2y — z 

..       ..      (2),  ..Jl:=f^ 

o 

Comparing  the  first  and  second  values  of  x,  we  have 

or,  40—4y—2z=Sl—y—3z 

or,  Sy—z=9     (4.) 

Comparing  the  first  and  third  values  of  x,  we  have 

^^    ^  44— 4//— 22 

20— 2//— 2==i ^ 

o 

or,  60— 6?/— 32=44— 4?/-22 

2y-f2=10     (5.) 

From  equations  (4)  and  (5),  the  values  of  y  and  z,  and  then  ar, 

may  be  found  by  the  rule.  Art.  159. 


144  RAY'S   ALGEBRA,    PART    FIRST. 

Solution  by  addition  and  subtraction. 

Multiplying  equation  (1)  by  2,  to  render  the  coefficient  of  a;  the 
same  as  in  equation  (2),  we  have 

2x+4ij+2z=40 
equation  (2)  is  2a;-[-  y+32=31 
by  subtracting,  'Si/—  2=   0     (4.) 

Next,  multiplying  equation  (1)  by  3,  to  render  the  coefficient  of 
X  the  same  as  in  equation  (3),  we  have 

3a:+6y+32=60 
equation  (3)  is  3a;+4?/+ 22^44 
by  subtracting,  2i/-\-  z=^Hj     (5.) 

3.V-   2=  9    (4.) 
by  adding,         5y    =    25 
y    =       b 
Then  10+2=16,  and  z=6. 
And      a:+10+6=20,  andar=4. 

Remark. — The  methods  of  eliminatiou  by  substitution  and  compari- 
son, when  there  are  more  than  two  unknown  quantities,  are  merely  an 
extension  of  the  rules  already  presented,  in  Articles  158  and  159;  there- 
fore, it  is  unnecessary  to  repeat  them  here.  When  the  number  of  unknown 
quantities  is  three  or  more,  and  particularly  when  each  of  the  unknown 
quantities  is  found  in  all  the  equations,  the  method  of  elimination  by  addi- 
tion*and  subtraction  is  generally  preferred;  we  shall,  therefore,  illustrate  it 
by  another  example. 

Let  it  be  required  to  find  the  value  of  each  of  the  unknown 
quantities  in  the  following  equations. 

r+2x-+3y +42=30     (1.) 
2y+3x+  7J+  2=15     (2.) 
Sv+  x+2y+32=23     (3.) 
4y+2x-?/+142=61     (4.) 
Let  us  first  eliminate  v:  this  may  be  done  by  making  the  coeffi- 
cient of  f,  in  one  of  the  equations,  the  same  as  in  the  other  three, 
and  then  subtracting. 

2y+4x+6y +82=60,  by  multiplying  equation  (1)  by  2. 
2?;+3a:+y+2=15  (2.) 

x-\r^y-{-lz-—4f)  (5.),  by  subtracting. 

3tJ+6x+%+ 122=90,  by  multiplying  equation  (1)  bv  3. 
3t;+x+2?/+32=23       (3.) 

5x+7y+92=:67       (6.),  by  subtracting. 
4?;+8x+12y+ 162=1 20,  by  multiplying  equation  (1)  by  4. 
4z?+2a:-y+142=   61      (4.) 

6a:+13//+22=  59     (7.),  by  subtracting. 


SIMPLE   EQUATIONS.  145 


Collecting  into  one  place,  the  new  equations  (5),  (6),  and  (7), 
we  find,  that  the  number  of  unknown  quantities,  as  well  as  the 
number  of  equations,  is  one  less. 

x+5y+7z=^5     (5.) 
5x-\-7y+9z=67     (6.) 
6x+lSy+2z=59    (7.) 
The  next  step  is  to  eliminate  x,  by  making  the  coefficient  of  x, 
in  one  of  the  equations,  the  same  as  in  each  of  the  others,  and 
then  subtracting. 

5x+25?/+35z=225,  by  multiplying  equation  (5)  by  5. 
bx+7y-{-9z=  67 
l8]/+26z=:l58    (8.) 
6a;+30?/+422=:270,  by  multiplying  equation  (5)  by  6. 
6x-^lSi/+2z=  59 

17y+40z=211     (9.) 
Bringing  together  equations  (8)  and  (9),  we  find,  that  the  num- 
ber of  equations,  as  well  as  of  unknown  quantities,  is  now  two 
less.  18i/+26z==158    (8.) 

17y+402=211     (9.) 
306^+4422=2686,  by  multiplying  equation  (8)  by  17. 
3Q6?/+720g=3798,  by  multiplying  equation  (9)  by  18. 
2783=1112 
z=       4 
Substituting  the  value  of  z,  in  equation  (9),  we  get 
17^+160=211 
17y=  51 
!/=     3. 
Substituting  the  values  of  y  and  z,  in  equation  (5),  we  get 
a;+ 15+28=45 
a;=2 
And  lastly,  substituting  the  values  of  x,  y,  and  z,  in  equation 
(1),  we  get  ^+4+9+16=30 

or,  v=^l. 
From  the  preceding  example,  we  derive  the 

GEA^ERAL   RULE, 

FOR   ELIMINATION    BY    ADDITION    AND   SUBTRACTION. 

1  St.  Combine  any  one  of  ike  equations  with  each  of  the  others,  so 
as  to  eliminate  the  same  unknown  quantity  ;  there  will  thus  arise  a 
new  class  of  equations,  containing  one  less  tinhiown  quantity. 

2d.  Combine  any  one  of  these  new  equations  with  each  of  the  others, 
so  as  to  eliminate  another  unknown  quantity ;  there  will  thus  aHse 
another  class  of  equations,  containing  two  less  unknown  quantities. 
13 


146 


RAY'S   ALGEBRA,   PART  FIRST. 


3d.  Cojitinue  this  series  of  operations  until  a  single  equation  is 
obtained,  containing  but  one  unknown  quantity,  from  which  its  value 
may  be  easily  found;  then,  by  going  back,  and  substituting  this  value 
in  the  derived  equations,  the  values  of  the  other  unknown  quantities 
may  be  readily  found. 

R  E  u  A  R  K. — When  the  number  of  unknown  quantities  in  each  equation, 
is  less  than  the  whole  number  of  unknown  quantities  involved,  the  method 
of  substitution  will  generally  be  found  the  shortest.  By  solving  several  of 
the  following  examples,  by  each  of  the  three  different  methods,  the  pupil 
will  be  able  to  appreciate  their  relative  excellence  in  different  cases. 


EXAMPLES, 

TO     BE     SOLVED    BY    EITHER     OF     THE     DIFFERENT     METHODS    OP 
ELIMINATION. 

1.  x+y^bOA Ans.  a;=18. 

x-^z=2S.  \ y=32. 

y+2=42.J 2=10. 

2.  2x-^by=  IQA      Ans.  a;=12. 

4a:+62=:108.  \ y=S. 

52+72/=106.J      2=10. 

3.  x+i/+2=26.  ^       Ans.  x=3. 

x-\-y—z=^—Q.  \      2/=7. 

x—y-\-z=\2.  J 2=16. 


4.  aJ+I=100: 


z  X  { ^^^'  «^=64. 

y+q=100;  2+2=100.    •   •   .     1  2/=72. 

I  2=84. 


5.  2x—y-\-z=9. 
a:-2?/+32=14. 

3x+4y— 22=7. 

6.  2x-3y+52=15. 

3a;+2y— 2=8. 

—x-h5y-f  22=21. 

^-  2^3^7-"^'^- 
|+M=31. 

|+M=32. 


Ans.  a:=:3. 

.  .    y=2. 

.     .       2=5. 

Ans.  x=2. 
.    .     2/=3. 

.    .      2=4. 


Ans.  a:=12. 

.   .>=30. 

,   .   .  2=42. 


X 

3" 

-|+.=3.  1 

6+4     3     ^'  1 

X 

2" 

-1+^5.  J 

Ans.  x=6. 
.   .    2/=4. 

.     .       13=3. 


SIMPLE   EQUATIONS.  147 


QUESTIOIVS   PRODUCING    EQUATIONS    COi\TAi:VIi\G     THREE    OR 
MORE    Ui\KNOWIV    QUANTITIES. 

Art.  163. — When  a  question  contains  three  or  more  unknown 
quantities,  equations  involving  them,  can  be  found  on  the  same 
principle  as  in  questions  containing  07ie  or  two  unknown  quanti- 
ties. (See  Articles  156  and  161.)  The  values  of  the  unknown 
quantities  may  then  be  found  by  either  of  the  three  methods  of 
elimination. 

Remark. — The  method  of  elimination  to  be  preferred,  will  depend  on 
the  manner  in  which  the  unknown  quantities  are  combined,  and  must  be 
left  to  the  judgment  of  the  pupil.  When  such  a  relation  exists  between  the 
different  unknown  quantities,  that  one  or  more  of  them  can  be  expressed 
directly  in  terms  of  another,  it  should  be  done,  as  this  generally  renders  the 
solution  more  simple. 

1.  A  person  has  3  ingots,  composed  of  3  different  metals  in  dif- 
ferent proportions ;  a  pound  of  the  first  contains  7  ounces  of  sil- 
ver, 3  of  copper,  and  6  of  tin  ;  a  pound  of  the  second  consists  of 
12  ounces  of  silver,  3  of  copper,  and  1  of  tin  ;  and  a  pound  of 
the  third,  of  4  ounces  of  silver,  7  of  copper,  and  5  of  tin.  How 
much  of  each  of  the  ingots  must  be  taken,  to  form  another  ingot 
of  1  pound  weight,  consisting  of  8  ounces  of  silver,  3|  of  copper, 
and  4|  of  tin? 

Let  X,  y,  2,  be  the  number  of  ounces  to  be  taken  of  the  3  ingots 
respectively. 

Then,  since  16  ounces  of  the  first  contain  7  ounces  of  silver, 
1  ounce  will  contain  jg  of  an  ounce  of  silver ;  and  hence,  x  ounces 

7x 
will  contain  ^^  ounces  of  silver. 
lb 

16 

4z 
ounces  of  silver ;  and  z  ounces  of  the  third  will  contain  yy.  ounces 

of  silver.     But,  by  the  question,  the  number  of  ounces  of  silver 
in  a  pound  of  the  new  ingot,  is  to  be  8,  hence 

16"^  16  ^16 

Or,  by  clearing  it  of  fractions, 

7x+12y+42=.128    (1.) 

Review. — 162.  What  is  the  general  rule  for  elimination  by  addition 
and  subtraction  ?  When  is  the  method  of  elimination  by  substitution  to  be 
preferred  to  this?  163.  Upon  what  principle  are  equations  formed,,  when 
a  question  contains  three  or  more  unknown  quantities  ?  When  should  we 
use  a  less  number  of  symbols  than  there  are  unknown  quantities  ? 


I4t8  RAY'S   ALGEBRA,    PART   FIRST. 


Reasoning  in  a  similar  manner  with  reference  to  the  copper  and 
the  tin,  we  have  the  two  following  equations : 
Sx+Su+7z=60    (2.) 
6x+  y+52=68     (3.) 
The  coefficient  of  y  being  the  simplest,  will  be  most  easily  elim- 
inated. 

If  we  multiply  the  second  equation  by  4,  and  take  the  first  equa- 
tion from  the  product,  the  result  is 

5xi-24z=n2    (4.) 
If  we  multiply  the  third  equation  by  3,  and  take  the  second 
from  the  product,  the  result  is 

15a:+8z=144     (5.) 
If  we  multiply  the  last  equation  by  3,  and  take  the  preceding 
equation  from  it,  the  result  is 

40x=320 
x=S 
Substituting  this  value  of  x  in  equation  (5),  we  have 
120+82=144 
z=S 
And  substituting  these  values  of  x  and  z,  in  equation  (3), 
48+y+ 15=68 
?/=5 
Hence,  the  new  ingot  will  contain  8  ounces  of  the  first,  5  of  the 
second,  and  3  of  the  third. 

2.  The  sums  of  three  numbers,  taken  two  and  two,  are  27,  32, 
and  35;  required  the  numbers.  Ans.  12,  15,  and  20. 

3.  The  sum  of  three  numbers  is  59  ;  ^  the  difi'erence  of  the 
first  and  second  is  5,  and  ^  the  difference  of  the  first  and  third  is 
9;  required  the  numbers.  Ans.  29,  19,  and  11. 

4.  There  are  three  numbers,  such  that  the  first,  with  ^  the  sec- 
ond, is  equal  to  14  ;  the  second,  with  ^  part  of  the  third,  is  equal 
to  18;  and  the  third,  with  {  part  of  the  first,  is  equal  to  20; 
required  the  numbers.  Ans.  8,  12,  and  18. 

5.  A  person  bought  three  silver  watches ;  the  price  of  the  first, 
with  :|  the  price  of  the  other  two,  was  25  dollars ;  the  price  of 
the  second,  with  -g  of  the  price  of  the  other  two,  was  26  dollars ; 
and  the  price  of  the  third,  with  ^  the  price  of  the  other  two,  was 
29  dollars  ;  required  the  price  of  each.         A.  $8,  $18,  and  $16. 

6.  Find  three  numbers,  such  that  the  first  with  -g  of  the  other 
two,  the  second  with  \  of  the  other  two,  and  the  third  with  i  of 
the  other  two,  shall  each  be  equal  to  25.        Ans.  13,  17,  and  19. 

7.  A  boy  bought  at  one  time  2  apples  and  5  pears,  for  12  cents; 
at  another,  3  pears  and  4  peaches,  for  18  cents ;  at  another,  4  pears 


,11 


A  dfrm 


SIMPLE   EQUATIONS.  149 


and  5  oranges,  for  28  cents ;  and  at  another,  5  peaches  and  6 
oranges,  for  39  cents ;  required  the  cost  of  each  kind  of  fruit. 
Ans.  Apples  1  cent,  pears  2,  peaches,  3,  oranges  4  cts.,  each. 

8.  A  and  B  together  possess  only  |  as  much  money  as  C  ;  B 
and  C  together,  have  6  times  as  much  as  A ;  and  B  has  680  dol- 
lars less  than  A  and  C  together  ;  how  much  has  each  ? 

Ans.  A  $200,  B  $360,  and  C  $840. 

9.  A,  B,  and  C  together,  have  1820  dollars;  if  B  give  A  200 
dollars,  then  A  will  have  160  dollars  more  than  B:  but  if  B 
receive  70  dollars  from  C,  they  will  both  have  the  same  sum  ;  how 
much  has  each  ?  Ans.  A  $400,  B  $040,  and  C  $780. 

10.  Three  persons,  A,  B,  and  C,  compare  their  money;  A  says 
to  B,  "Give  me  700  dollars,  and  I  shall  have  twice  as  much  as  you 
will  have  left."  B  says  to  C,  "Give  me  1400  dollars,  and  I  shall 
have  three  times  as  much  as  you  will  have  left."  And  C  says  to 
A,  "Give  me  420  dollars,  and  then  I  shall  have  five  times  as  much 
as  you  will  have  left."     How  much  has  each  ? 

Ans.  A  $980,  B  $1540,  and  C  $2380. 

1 1.  A  certain  number  is  expressed  by  three  figures,  and  the  sum 
of  the  figures  is  1 1 ;  the  figure  in  the  place  of  units,  is  double  that 
in  the  place  of  hundreds  ;  and  if  297  be  added  to  the  number,  its 
figures  will  be  inverted  ;  required  the  number.  Ans.  326. 

12.  Three  persons.  A,  B,  and  C,  together,  have  2000  dollars; 
if  A  gives  B  200  dollars,  then  B  will  have  100  dollars  more  than 
C;  but,  if  B  gives  A  100  dollars,  then  B  will  have  only  |  as  much 
as  C  ;  required  the  sum  possessed  by  each. 

Ans.  A  $500,  B  $700,  and  C  $800. 

13.  There  are  three  numbers  whose  sum  is  83  ;  if,  from  the 
first  and  second  you  subtract  7,  the  remainders  are  as  5  to  3 ;  but 
if  from  the  second  and  third,  you  subtract  3,  the  remainders  are 
to  each  other  as  11  to  9  ;  required  the  numbers.      A.  37,  25,  21. 

14.  Divide  180  dollars  between  three  persons.  A,  B,  and  C,  so 
that  twice  A's  share  plus  80  dollars,  three  times  B's  share,  plus 
40  dollars,  and  four  times  C's  share  plus  20  dollars,  may  be  all 
equal  to  each  other.  Ans.  A  $70,  B  $60,  and  C  $50." 

15.  There  are  three  numbers  whose  sum  is  78 ;  -3-  of  the  first  is 
to  I  of  the  second,  as  1  to  2  ;  also,  -|  of  the  second  is  to  ^  of  the 
third,  as  2  to  3  ;  what  are  the  numbers?         Ans.  9,  24,  and  45. 

16.  A,  B,  and  C,  have  a  sum  of  money ;  A's  share  exceeds  4  of 
the  shares  of  B  and  C,  by  30  dollars;  B's  share  exceeds  |-  of  the 
shares  of  A  and  C,  by  30  dollars ;  and  C's  share  exceeds  |  of  the 
shares  of  A  and  B,  by  30  dollars;  what  is  the  share  of  each? 

Ans.  A's  $150,  B's  $120,  and  C's  $90. 


150  RAY'S   ALGEBRA,    PART   FIRST. 

17.  If  A  and  B  can  perform  a  certain  work  in  12  days,  A  and 
C  in  15  days,  and  B  and  C  in  20  days,  in  what  time  could  each 
do  it  alone  ?  Ans.  A  20,  B  30,  and  C  60  days. 

18.  A  number,  expressed  by  three  figures,  when  divided  by  the 
sum  of  the  figures  plus  9,  gives  a  quotient  19  ;  also,  the  middle 
figure  is  equal  to  half  the  sum  of  the  first  and  third ;  and,  if  198 
be  added  to  the  number,  we  obtain  a  number  with  the  same  figures 
in  an  inverted  order;  what  is  the  number?  Ans.  456. 

19.  A  farmer  mixes  barley  at  28  cents,  with  rye  at  36,  and 
wheat  at  48  cents  per  bushel,  so  that  the  whole  is  100  bushels, 
and  worth  40  cents  per  bushel.  Had  he  put  twice  as  much  rye, 
and  10  bushels  more  of  wheat,  the  whole  would  have  been  worth 
exactly  the  same  per  bushel;  how  much  of  each  kind  was  there? 

Ans.  Barley  28,  rye  20,  and  wheat  52  bushels. 

20.  A,  B,  and  C,  in  a  hunting  excursion,  killed  96  birds,  w^hich 
they  wish  to  share  equally ;  in  order  to  do  this.  A,  who  has  the 
most,  gives  to  B  and  C  as  many  as  they  already  had ;  next,  B  gives 
to  A  and  C  as  many  as  they  had  after  the  first  division  ;  and 
lastly,  C  gives  to  A  and  B  as  many  as  they  both  had  after  the 
second  division;  it  was  then  found,  that  each  had  the  same  num- 
ber ;  how  many  had  each  at  first?        Ans.  A  52,  B  28,  and  C  16. 


CHAPTER  V. 

SUPPLEMENT   TO  EQUATIONS  OF  THE  FIRST  DEGREE. 

GENERALIZATION. 

Art.  164. — Equations  are  termed  literal,  when  the  known 
quantities  are  represented,  either  entirely  or  partly,  by  letters. 
Quantities  represented  by  letters,  are  termed  general  values — be- 
cause, by  giving  particular  values  to  the  letters,  the  solution  of  one 
problem,  furnishes  a  general  solution  to  all  others  of  the  same  kind. 

The  answer  to  a  problem,  when  the  known  quantities  are  repre- 
sented by  letters,  is  termed  n,  formula;  and  a  formula,  expressed 
in  ordinary  language,  furnishes  a  ride. 

By  the  application  of  Algebra  to  the  solution  of  general  ques- 
tions, a  great  number  of  useful  and  interesting  truths  and  rules 
may  be  established.  We  shall  now  proceed  to  illustrate  this  sub- 
ject, by  a  few  examples. 

Art.  165. — 1.  Let  it  be  required  to  find  a  number,  which  being 
divided  by  3,  and  by  5,  the  sum  of  the  quotients  will  be  16. 


GENERALIZATION.  151 

Let  ar=  the  number;  then  ^+^=16. 
3    5 

5x+3a:=16Xl5 

8a:=16X15 
x=  2X15=30. 

2.  Again,  let  it  be  required  to  find  another  number,  which  being 
divided  by  4,  and  by  7,  the  sum  of  the  quotients  will  be  1 1 . 

By  proceeding,  as  in  the  preceding  question,  we  find  the  num- 
ber to  be  28. 

Instead,  however,  of  solving  every  example  of  the  same  kind 
separately,  we  may  give  a  general  solution,  that  will  embrace  all 
the  particular  questions.     Thus: 

3.  Let  it  be  required  to  find  a  number,  which  being  divided  by 
two  given  numbers,  a  and  b,  the  sum  of  the  quotients  may  bo 
equal  to  another  given  number,  c.  ^ 

X      X 

Let  x=  the  number ;  then  — \-t=c. 
a    o 

bx-\-ax=abc 

{a-{-b)x=abc 

abc 

The  answer  to  this  question  is  termed  a  formula  ;  it  shows,  that 
the  required  number  is  equal  to  the  continued  product  of  a,  b,  and 
c,  divided  by  the  sum  of  a  and  b.  Or,  it  may  be  expressed  in 
ordinary  language,  thus :  Multiply  together  the  three  given  numberSy 
and  divide  the  product  by  the  sum  of  the  divisors;  the  result  will  be 
the  required  nurnber. 

The  pupil  may  test  the  accuracy  of  this  rule,  by  solving  the 
following  examples,  and  verifying  the  results. 

4.  Find  a  number,  which  being  divided  by  3,  and  by  7,  the  sum 
of  the  quotients  may  be  20.  Ans.  42. 

5.  Find  a  number,  which  being  divided  by  \  and  \,  the  sum  of 
the  quotients  may  be  1.  Ans.  4- 

Art.  166. — 1.  The  sum  of  500  dollars  is  to  be  divided  between 
two  persons,  A  and  B,  so  that  A  may  have  50  dollars  less  than  B. 

Ans.  A  $225,  B  $275. 
To  make  this  question  general,  let  it  be  stated  as  follows: 

Revieav. — 164.  When  are  equations  termed  literal ?  When  are  quan- 
tities termed  general  ?  When  is  the  answer  to  a  problem  termed  a  formula  ? 
What  is  a  formula  called,  when  expressed  in  ordinary  language  ?  165.  Ex- 
ample 3.  What  is  the  answer  to  this  question,  expressed  in  ordinary 
language  ? 


152  RAY'S   ALGEBRA,    PART   FIRST. 

2.  To  divide  a  given  number,  a,  into  two  such  parts,  that  their 
difference  shall  be  b.     Or  thus : 

The  sum  of  two  numbers  is  a,  and  their  difference  b ;  required 
the  numbers. 

Let  x=  the  greater  number,  and  y=  the  less. 
Then  x-{-y=:a 
And  X — ?/=6 
By  addition,  2x=a-\-b 

a-\-b a      b 

By  subtraction,  2y=^a—b 

a — b a      b 

y~^r'~2~2' 

This  formula,  when  expressed  in  ordinary  language,  gives  the 

RULE, 

FOR   FINDING    TWO   QUANTITIES,    WHEN    THEIR     SUM    AND    DIFFERENCE 
ARE    GIVEN. 

To  find  the  greater,  add  half  the  difference  to  half  the  sum.  To 
find  the  less,  subtract  half  the  difference  from  half  the  sum. 

Let  the  learner  test  the  accuracy  of  the  rule,  by  finding  two 
numbers,  such  that  their  sum  shall  be  equal  to  the  first  number 
in  each  of  the  following  examples,  and  their  difference  equal  to 
the  second. 

3.  Sum  200,  difference  50 Ans.  125,  75. 

4.  Sum  100,  difference  25 Ans.  62^,  37^. 

5.  Sum  15,  difference  10 -Ans.  12^,  2|. 

6.  Sum  5i,  difference  | Ans.  3|,  2|. 

Art.  167. — 1.  A  can  perform  a  certain  piece  of  work  in  3  days, 

and  B  in  4  days ;  in  what  time  can  they  both  together  do  it  ? 

Ans.  If  days. 
To  make  this  question  general,  let  it  be  stated  thus : 
2.  A  can  perform  a  certain  piece  of  work  in  m  days,  and  B  can 

do  it  in  11  days ;  in  how  many  days  can  they  both  together  do  it  ? 
Let  ic=  the  number  of  days  in  which  they  can  both  do  it. 

Then  -=  the  part  of  the  work  which  both  can  do  in  one  day. 

Also,  if  A  can  do  the  M'ork  in  m  days,  he  can  do  —  part  of  it  in 

one  day.     And,  if  B  can  do  the  work  in  n  days,  he  can  do  -  part 
of  it  in  one  day.     Hence,  the  part  of  the  Avork  which  both  can  do 

in  one  day,  is  represented  by  — | — ,  and  also  by  -. 
'    m      a  X 


GENERALIZATION.  153 


Therefore, —+-=-. 

m     n     X 

nx-\-mx-=zmn 

mn 

x= — — . 

This  result,  expressed  in  ordinary  language,  gives  the  following 

RULE. 

Divide  the  product  of  the  numbers  expressing  the  time  in  ivhich 
each  can  perform  the  work  by  their  sum ;  the  quotient  will  be  the 
time  in  which  they  can  jointly  perform  it. 

The  question  can  be  made  more  general,  by  expressing  it  thus : 
An  agent,  A,  can  produce  a  certain  effect,  e,  in  a  time,  t-,  another 
agent,  B,  can  produce  the  same  effect,  in  a  time,  t';  in  what  time 
can  they  both  do  it?  Both  the  result  and  the  rule  would  be  the 
same  as  that  already  given. 

The  following  examples  will  illustrate  the  rule. 

3.  A  cistern  is  filled  by  one  pipe  in  6,  and  by  another  in  9  hours ; 
in  what  time  will  it  be  filled  by  both  together?  A.  3|  hrs. 

4.  One  man  can  drink  a  keg  of  cider  in  5  days,  and  another  in 
7  days ;  in  what  time  can  both  together  drink  it?        A.  U>\h  dys. 

Art.  16S. — Let  it  be  required  to  find  a  rule  for  dividing  the  gain 
or  loss  in  a  partnership,  or,  as  it  is  generally  termed,  fellowship. 
First,  take  a  particular  question. 

1.  A,  B,  and  C,  engage  in  trade,  and  put  in  stock  in  the  follow- 
ing proportions:  A  put  in  3  dollars,  as  often  as  B  put  in  4,  and  as 
often  as  C  put  in  5  dollars.  Their  gains  amounted  to  60  dollars ; 
required  the  share  of  each,  the  gains  being  divided  in  proportion 
to  the  stock  put  in. 

Let  3a:=  A's  share  of  the  gain,  then  4a;=  B's,  and  bx=  C's. 
(See  Example  24,  page  126.) 
Then  3x+4x+5x=-60 
or,  12x=60 
a:=  5 
3x=15,  A^s  share, 
4x=20,  B's      " 
5x=25,  C's      " 

2.  To  make  this  question  general,  suppose  A  puts  in  m  dollars, 
as  often  as  B  puts  in  n  dollars,  and  as  often  as  C  puts  in  r  dolhirs; 
and  that  they  gain  c  dollars.     To  find  the  share  of  each. 

Review. — 166.  By  what  rule  do  you  find  two  quantities,  when  their 
sum  and  difi'erence  are  given?  167.  When  the  times  are  given,  in  which 
each  of  two  agents  can  produce  a  certain  eflfect,  how  is  the  time  found  iu 
which  they  can  jointly  produce  it? 


154  RAY'S   ALGEBRA,    PART  FIRST. 

Let  the  share  of  A  be  denoted  by  mx,  then  nx=  B's,  and  ra:= 

C's  share.     Then  mx-{-nx-{-rx=c 

c 
x= 

m-\-7i-\-r 

mx=mX  — \ — r-= — \ — r- 

c  nc 

nx=7iX. 


rx=r'X 


m-{-n-\-r    m-\-n-{-r 
c  re 


m-j-u-f-r  7n-\-n-\-r 
By  examining  these  formula,  we  see  that  the  whole  gain,  c,  is 
divided  by  m-]-n-\-r,  the  sum  of  the  proportions  of  stock  furnished 
by  all  the  partners,  and  that  this  quotient  is  multiplied  by  m,  «, 
and  r,  each  one's  respective  proportion,  to  obtain  his  share  of  tho 
gain. 

If  c  had  represented  loss,  instead  of  gain,  the  same  solution 
would  have  applied.  Hence,  to  find  each  partner's  share  of  the 
gain  or  loss,  we  have  the  following 

RULE. 

Divide  the  wJiole  gain  or  loss  by  the  sum  of  the  proportions  of 
stock,  and  midiiply  the  quotient  by  each  partner's  proportion,  to 
obtain  his  respective  share. 

When  the  times  in  which  the  respective  stocks  are  employed 
are  different,  it  becomes  necessary  to  reduce  them  to  the  same 
time,  to  ascertain  what  proportion  they  bear  to  each  other. 

Thus,  if  A  have  3  dollars  in  trade  4  months,  and  B  2  dollars  5 
months,  we  see,  that  3  dollars  for  4  months,  are  the  same  as  12 
dollars  for  1  month  ;  and  2  dollars  for  5  months,  are  the  same  as 
10  dollars  for  one  month.  Therefore,  in  this  case,  the  gain  or  loss 
must  be  divided  in  the  proportion  of  12  to  10  ;  that  is,  in  propor- 
tion to  the  product  of  the  stocks  by  the  times  in  which  they  were 
employed.     Hence,  when  time  in  fellowship  is  considered,  we  have 

the  following 

RULE. 

Midtiply  each  mail's  stock  by  the  time  during  which  it  wa^  em- 
ployed ;  and  then,  according  to  the  preceding  rule,  divide  the  gain 
or  loss  in  proportion  to  these  products. 

3.  A,  B,  and  C  engaged  in  trade  ;  A  put  in  200  dollars,  B  300, 
and  C  700;  they  lost  60  dollars;  what  was  each  man's  sliaro? 

Ans.  A's  loss  $10,  B's  $15,  and  C's  $35. 

11  E  VIE  w. — 168.  How  is  the  gain  or  loss  in  fellowship  found,  when  tho 
times  in  which  the  stock  is  employed  are  the  same  ?  How  is  it  found,  when 
the  times  are  different? 


GENERALIZATION.  155 

Since  the  sums  engaged,  evidently  are  to  each  other,  as  2,  3, 
and  7,  we  may  either  use  these  numbers,  or  those  representing  the 
stock. 

4.  In  a  trading  expedition  A  put  in  200  dollars  3  months,  B 

150  dollars  for  5  months,  and  C  100  dollars  for  8  months;  they 

gained  215  dollars ;  what  was  each  man's  share  of  the  gain  ? 

Ans.  A's  share  $60,  B's  $75,  and  C's  " 

Art.  169. — 1 .  Two  men,  A  and  B,  can  perform  a  certain  piece 
of  work  in  a  days,  A  and  C  in  6  days,  and  B  and  C  in  c  days ;  in 
what  time  could  each  one,  alone,  perform  it?  and,  in  what  time 
could  they  perform  it,  all  working  together  ? 

Let  X,  y,  and  z  represent  the  days  in  which  A,  B,  and  C  can 
respectively  do  it. 

Then  -,  -,  and  -,  represent  the  parts  of  the  work  which  A,  B, 
X  y         z       ^ 

and  C  can  each  do  in  1  day. 

Since  A  and  B  can  do  it  in  a  days,  they  do  -  part  of  it  in  1  day. 

But,  — I —  represents  the  part  of  the  work  which  A  and  B  can  do 

X    y 
in  one  day.     Hence, 

-J — =-     (1.)     and  reasoning  in  a  similar  manner,  we  have 

x^  y    a     ^    ' 

2    2    2     111 

-4---1— =-+--1 — ,  by  adding  the  three  equations  together. 
X    y    z     a     0    c 

1      1  +  1  _  1  =^£^^"-^  by  subtracting  (3)  from  (4). 
X    2a    2b     2c         2aoc 

or,  x{ac-]-bc—ah)=2abc,  by  clearing  of  fractions. 

In  a  similar  manner,  by  subtracting  equation 


ac-\-bc — ab 

.    .  '^abc 

(2)  from  (4),  and  reducmg,  we  find  2/= -^^,^^c* 

Also,  in  the  same  manner,  z  is  found  =^^  ,  ^^._^^> 

Since  -+-+-,  or  J-hoT+o-»  represents  the  part  all  can  do  in 
X    y    z        t^a    -w"    '^^ 


156  RAY'S   ALGEBRA,   PART    FIRST. 

one  day;  if  we  divide  1  by  I  ^^ — H9T+9-  )  >  the  quotient,  which 

is  -7-^ n-»  will  represent  the  number  of  days  in  which  all  can 

ab+ac+bc  ^  '^ 

perform  it. 

Art.  I'yO. — In  the  solution  of  questions,  it  is  sometimes  neces- 
sary to  use  general  values  for  particular  quantities,  to  ascertain 
the  relation  which  they  bear  to  each  other ;  as  in  the  following 
problem. 

If  4  acres  pasture  40  sheep  4  weeks,  and  8  acres  pasture  56 
sheep  10  weeks,  how  many  sheep  will  20  acres  pasture  50  weeks, 
the  grass  growing  uniformly  all  the  time  ? 

The  chief  difficulty  in  solving  this  question,  consists  in  ascer- 
taining the  relation  that  exists  between  the  original  quantity  of 
grass  on  an  acre,  and  the  groAvth  on  each  acre  in  one  week. 

Let  m=:  the  quantity  on  an  acre  when  the  pasturage  began,  and 
n^^  the  growth  on  1  acre  in  1  week;  m  and  n  representing  pounds, 
or  any  other  measure  of  the  quantity  of  grass.  , 

Then  4n=  the  growth  on  1  acre  in  4  weeks. 

And  16n=  the  growth  on  4  acres  in  4  weeks. 

Also,  4m+16»=  the  whole  amount  of  grass  on  4  acres  in  4 
weeks. 

If  40  sheep  eat  47n-\-l6n  in  4  weeks,  then  40  sheep  eat 

4m+16w         ,  A     '  1 
=OT+4n  in  one  week. 

.     ,  ,    ,  ,    m+4n     m  ^   n   .  . 

And  1  sheep  eats  -^ri — — Jn"^]?)  ^^  ^^^  week. 

Again,  Sm-\-807i=  the  whole  amount  of  grass  on  8  acres  in  10 
weeks. 

If  56  sheep  eat  8;/i+80/i  in  10  weeks, 

Then  56  sheep  eat  y>Y+8?^  in  1  week. 
And  1  sheep  eats  5(jo~^56^70^7  ^^     "^ 

Henc^,  4o+r0^70^  f 
Or,  7wi+28?i=4m+40w 
3m=127i 
w=4n 
or  w=|w ;  hence,  the  growth  on  one  acre  in  1  week,  is 
equal  to  \  of  the  original  quantity  on  an  acre. 

Then,  1  sheep,  m  1  week,  eats  ^^+--^=--^r-^-. 


GENERALIZATION.  157 


And  1  sheep,  in  50  weeks,  eats  ^X50=-4y-. 

20  acres  have  an  original  quantity  of  grass,  denoted  by  20m. 
The  growth  of  1  acre  in  1  week  being  ^m,  in  50  weeks,  it  will 

be  —J—.     And  the  growth  of  20  acres,  in  50  weeks,  will  be 

^-X20==250;>^ 

Then  207?t-|-250m=270m,  the  whole  amount  of  grass  on  20 
acres  in  8  weeks. 

Then  270m-. — --=— — =108,  the  number  of  sheep  required. 

GENERAL  PROBLEMS. 

1.  Divide  the  number  a  into  two  parts,  so  that  one  of  them  shall 

be  71  times  the  other.  ,  na         .      a 

Ans.  ~—^  and  — -.. 
?t+ 1  n~\- 1 

2.  Divide  the  number  a  into  two  parts,  so  that  7n  times  one  part 

shall  be  equal  to  n  times  the  other.  .  na         .     ma 

^  Ans.  — —  and  — —  . 

m-i-u  m-\-n 

3.  Divide  the  number  a  into  two  parts,  so  that  when  the  first  is 

multiplied  by  m,  and  the  second  by  n,  the  sum  of  the  products  may 

be  equal  to  b.  ,        b — 7ia       ,  ma—b 

^  Ans. and . 

7)1 — a  m — 71 

4.  Find  a  number,  which  being  divided  by  in,  and  by  n,  the  sum 
of  the  quotients  shall  be  equal  to  a.  .         mna 

m-\-n' 

5.  Divide  a  into  three  such  parts,  that  the  second  shall  be  m, 
and  the  third  n  times  the  first. 

.              a               7na  .        7ia 

Ans.  ^i— ; — ,  -,— ; — ,  and 


\-\-7n-\-7i'    iH-w+n'  \-{-m-{-n 

6.  Divide  a  into  two  such  parts,  that  one  of  them  being  divided 

by  6,  and  the  other  by  c,  the  sum  of  the  quotients  shall  be  equal 

to  d  .        bia — cd)       -  dbd — a) 

Ans.  -\ and  — , '. 

b—c  b — c 

7.  What  number  must  be  added  to  a  and  6,  so  that  the  sums 

shall  be  to  each  other  as  m  to  w  ?  .        w6 — 7ia 

Ans.  . 

71 — m 

8.  What  number  must  be  subtracted  from  a  and  6,  so  that  the 

differences  shall  be  to  each  other  as  m  to  w  ?  .        na — mb 

Ans.  . 

71 — m 

9.  What  number  must  be  added  to  a,  and  subtracted  from  b,  that 
the  sum  may  be  to  the  difference  as  w  to  7i  ?  ^        w6 — na 

"■    in-{-n 


158  EAY'S   ALGEBRA,    PART    FIRST. 

10.  After  paying  away  —  and  —  of  my  money,  I  had  a  dollars 
left ;  how  many  dollars  had  I  at  first?  .  mna 


mn — m — n 


1 1 .  What  quantity  is  that  of  which  the  —  part,  diminished  by 

the  -  part,  is  equal  to  a  ?  Ans.  ~ . 

q  mq — np 

12.  A  certain  number  of  persons  paid  for  the  use  of  a  boat,  for 
a  pleasure  excursion,  a  cents  each  ;  but,  if  there  had  been  b  per- 
sons less,  each  would  have  had  to  pay  c  cents ;  how  many  persons 

were  there?  ,  be 

Ans.  — — . 
c — a 

13.  A  person  gave  some  poor  persons  a  cents  a  piece,  and  had  6 

cents  left ;  but,  if  he  had  given  them  c  cents  a  piece,  he  would 

have  had  d  cents  left ;  hoAV  many  persons  were  there  ?   .         d — b 

Ans.  . 

a — c 

14.  A  farmer  mixes  oats  at  a  cents  per  bushel,  with  rye  at  b 

cents  per  bushel,  so  that  a  bushel  of  the  mixture  is  worth  c  cents ; 

how  many  bushels  of  each  will  n  bushels  of  the  mixture  contain  ? 

.        n{c — b)        ,  7i(a — c) 

Ans.  — !^ ~  and  — ^^ — —'. 

a — b  a — b 

15.  A  person  borrowed  as  much  money  as  he  had  in  his  purse, 

and  then  spent  a  cents ;  again,  he  borrowed  as  much  as  he  had  in 

his  purse,  after  which  he  spent  a  cents ;  he  borrowed  and  spent, 

in  the  same  manner,  a  third  and  fourth  time,  after  which,  he  had 

nothing  left;  how  much  had  he  at  first?  .         15a 

Ans.  ^g. 

16.  A  person  has  2  kinds  of  coin  ;  it  takes  a  pieces  of  the  first, 

and  b  pieces  of  the  second,  to  make  one  dollar ;  how  many  pieces 

of  each  kind  must  be  taken,  so  that  c  pieces  may  be  equivalent  to 

a  dollar?  .        aib — c)       ,  b(c—aX 

Ans.  —. and  —. -. 

b—a  b—a 

Art.  171. — It  sometimes  happens  in  the  solution  of  an  equa- 
tion of  the  first  degree,  that  the  second  or  some  higher  power  of 
the  unknown  quantity  occurs  ;  but,  in  such  a  manner,  that  it  is 
easily  removed,  or  made  to  disappear,  so  that  the  equation  can  be 
solved  in  the  usual  manner.  The  following  are  examples  of  equa- 
tions and  problems  belonging  to  this  class. 

1.  Given  2x^+8a:=llx^ — lOx,  to  find  the  value  of  x. 
By  dividing  each  side  by  x,  we  have 

2x+8=llx-10,  from  which  x=2. 

2.  Given  (4+aj)(3+x)-6(10 -x)=a;(7+a;),  to  find  x. 


NEGATIVE    SOLUTIONS.  159 

Performing  the  operations  indicated,  we  have 

Omitting  the  quantities  on  each  side  which  are  equal,  we  have 
12— 60+6x=0,  from  which  x=^8. 

3.  Sx^—8x=24x-bx-' Ans.  x^4. 

4.  40x2-6r^— 16a;2=:120x2— 14x3 Ans.  x=12. 

5.  3ax3— 10ax2=8ax2+aar' Ans.  a:=<). 

2x^      x^ 

6.  ic'^+-q o~^ Ans.  x=^. 

^    6x+13      3x+5      2x  ,  „^ 

^-  -15- -5^=25^5 Ans.x=20. 

8.  {a-\-x){b-i-x)—a{c—b)=:x{b+x) Ans.  x=c— 2/>. 

„          ,  ,,     a{x'-]-c')  .  a{c'—b') 

^'  ^+^-^- ^^''^=^W' 

10.  x^a^^+c=^^±^±^ Ans.x=.^-!=«^. 

a^b—c+x  a+6 

11.  The  difference  between  two  numbers  is  2,  and  their  product 
is  8  greater  than  the  square  of  the  less  ;  what  are  the  numbers? 

Ans.  4  ana  t>. 

12.  It  is  required  to  divide  the  number  a  into  two   such  parts, 

that  the  difference  of  their  squares  may  be  c. 

.        a}—c       ,  a'+c 
Ans.  —^ —  and  — r —  . 

13.  If  a  certain  book  contained  5  more  pages,  with  10  more 
lines  on  a  page,  the  number  of  lines  would  be  increased  450 ;  but 
if  it  contained  10  pages  less,  with  5  lines  less  on  a  page,  the  Avhole 
number  of  lines  would  be  diminished  450.  Required  the  number 
of  pages,  and  the  number  of  lines  on  a  page. 

Ans.  20  pages,  and  40  lines  on  a  page. 

NEGATIVE   SOLUTIONS. 

Art.  172.— It  has  been  stated  already  (Art.  23),  that  when  a 
quantity  has  no  sign  prefixed,  the  sign  plus  is  understood ;  and 
also  (Art.  64),  that  all  numbers  or  quantities  are  regarded  as  posi- 
tive, unless  they  are  otherwise  designated.  Hence,  in  all  prob- 
lems, it  is  understood,  that  the  results  are  required  in  positive 
numbers.  It  sometimes  happens,  however,  that  the  value  of  the 
unknown  quantity  in  the  solution  of  a  problem,  is  found  to  bo 
minus.  Such  a  result  is  termed  a  negative  solution.  AVe  shall  now 
examine  a  question  of  this  kind. 

1.  What  number  must  be  added  to  the  number  5,  that  the  sum 
shall  be  equal  to  3  ? 

Let  x=  the  number. 

Then  5+x=3. 

And  a;=3-5=— 2. 


100  RAY'S   ALGEBRA,    PART    FIRST. 

Now,  —2  added  to  5,  according  to  the  rule  for  Algebraic  Addi- 
tion, gives  a  sum  equal  to  3  ;  thus,  5-}-(— 2)=3.  The  result,  — 2, 
is  said  to  satisfy  the  question  in  an  algebraic  sense;  but  the  prob- 
lem is  evidently  impossible  in  an  arithmetical  sense,  since  any  posi- 
tive number  added  to  5,  must  increase,  instead  of  diminishing  it ; 
and  this  impossibility  is  shown,  by  the  result  being  negative,  in- 
stead of  positive.  Since  adding  — 2,  is  the  same  as  subtracting 
-|-2  (Art.  61),  the  result  is  the  answer  to  the  following  question: 
What  number  must  be  subtracted  from  5,  that  the  remainder  may 
be  equal  to  3  ? 

Let  the  question  now  be  made  general,  thus  : 

What  number  must  be  added  to  the  number  a,  that  the  sum 
shall  be  equal  to  6  ? 

Let  x=  the  number. 

Then  a+a;=6. 

And  x=b — a. 

Now,  since  a-\-{b—a)=b,  this  value  of  x  will  always  satisfy  the 
question  in  an  algebraic  sense. 

While  5  is  greater  than  a,  the  value  of  x  will  be  positive,  and, 
whatever  values  are  given  to  b  and  a,  the  question  will  be  consist- 
ent, and  can  be  answered  in  an  arithmetical  sense.  Thus,  if  6=10, 
and  «=8,  then  x^=2. 

But  if  b  becomes  less  than  a,  the  value  of  x  will  be  negative; 
and  whatever  values  are  given  to  b  and  a,  the  result  obtained,  will 
satisfy  the  question  in  its  «?^e6?mc,  but  not  in  its  arithmetical  sense. 

Thus,  if  6=5,  and  a=8,  then  x=— 3.  Now  8+(--3)=5;  that 
is,  if  we  subtracts  from  8,  the  remainder  is  5.  We  thus  see,  that 
w^hen  a  becomes  greater  than  b,  the  question,  to  be  consistent, 
should  read,  What  number  must  be  subtracted  from  the  number  a, 
that  the  remainder  shall  be  equal  to  6?     From  this  we  see, 

1st.  That  a  negative  solution  indicates  some  inconsistency  or  ab- 
surdity, in  the  question  from  which  the  equation  was  derived. 

2d.  When  a  negative  solution  is  obtained,  the  question,  to  which  it 
is  the  answer,  may  be  so  7nodiJied  as  to  be  consistent. 

Let  the  pupil  now  read,  carefully,  the  "  Observations  on  Addi- 
tion AND  Subtraction,'*  page  43,  and  then  modify  the  following 
questions,  so  that  they  shall  be  consistent,  and  the  results  true  in 
an  arithmetical  sense. 

2.  What  number  must  be  subtracted  from  20,  that  the  remainder 
shall  be  25?     (x=-5.) 

R  E  V I E  w. — 172.  What  is  a  negative  solution  ?  When  is  a  result  said  to 
eatisfy  a  question  in  an  algebraic  sense  ?  In  an  arithmetical  senso  ?  What 
does  a  negative  solution  indicate  ?    - 


DISCUSSION   OF    PROBLEMS.  101 


3.  What  number  must  be  added  to  11,  that  the  sum  beiug  mul- 
tiplied by  5,  the  product  shall  be  40?     (a-— — 3.) 

4.  What  number  is  that,  of  %Yhieh  the  |  exceeds  the  f  by  3  ? 

(x-=— 36.) 

5.  A  father,  whose  age  is  45  years,  has  a  son,  aged  15 ;  inliow 
many  years  will  the  son  be  \  as  old  as  his  father?     (x=^-  -5.) 

DISCUSSION    OF    PROBLEMS. 

Art.  173. — When  a  question  has  been  solved  in  a  general  man- 
ner,  that  is,  by  representing  the  known  quantities  by  letters,  we 
may  inquire  what  values  the  results  will  have,  when  particular 
suppositions  are  made  with  regard  to  the  known  quantities.  The 
determination  of  these  values,  and  the  examination  of  the  various 
results  which  we  obtain,  constitute  what  is  termed  the  discussion 
of  the  problem. 

The  various  forms  which  the  value  of  the  unknown  quantity  may 
assume,  are  shown  in  the  discussion  of  the  following  question. 

1.  After  subtracting  h  from  a,  what  number,  multiplied  by  the 
remainder,  will  give  a  product  equal  to  c? 

Let  x=  the  number. 

Then  [a — h)x=c. 

c 

x= -. 

a—o 

Now,  this  result  may  have  five  different  forms,  depending  on  the 

values  of  a,  b,  and  c. 

Note.  —  In  the  following  forms,  A  denotes  merely  some  quantity. 

1st.  When  b  is  less  than  a.  This  gives  positive  values,  of  the 
form  -{-A. 

2d.  When  b  is  greater  than  a.  This  gives  negative  values,  of 
the  form  — A. 

3d.  When  b  is  equal  to  a.     This  gives  values  of  the  form  ^. 

4th.  Where  c  is  0,  and  b  either  greater  or  less  than  a.  This 
gives  values  of  the  form  £. 

5th.  When  b  is  equal  to  a,  and  c  is  equal  to  0.  This  gives 
values  of  the  form  g. 

We  shall  examine  each  of  these  in  succession. 

L  When  b  is  less  than  a. 

In  this  case,  a—b  is  positive,  and  the  value  of  x  is  positive. 
To  illustrate  this  form,  let  a— 8,  6=3,  and  c— 20,  then  x=4. 

Review. — 172.  When  a  negative  solution  is  obtained,  how  may  the 
question,  to  which  it  is  the  answer,  be  modified  ?  173.  What  do  you  under- 
Gtand  by  the  discussion  of  a  problem?  The  expression  c  divided  by  a— 6, 
may  have  how  manv  forms?     Name  these  different  forms. 

u 


162  RAT'S   ALGEBRA,    PART   FIRST. 

II.  When  b  is  greater  than  a. 

In  this  case,  a — b  is  a  negative  quantity,  and  the  value  of  x 
will  be  negative.  This  evidently  should  be  so,  since  minus  mul- 
tiplied by  minus  produces  plus;  that  is,  if  a — 6  is  minus,  x  must 
be  minus,  in  order  that  their  product  shall  be  equal  to  c,  a  posi- 
tive quantity.  To  illustrate  this  case  by  numbers,  let  a=2,  6=5, 
and  c=12;  then,  a— 6=— 3,  «;=:— 4,  and  — 3X— 4=12. 

III.  When  b  is  equal  to  a. 

Q 

In  this  case  x  becomes  equal  to  t^.     We  must  now  inquire,  what 
is  the  value  of  a  fraction  when  the  denominator  is  zero. 
1st.  Suppose  the  denominator  1,  then  t=c. 

2d.    Suppose  the  denominator  jo>  then  y=10c. 

3d.    Suppose  the  denominator  jJq,  then         -=10Qc. 

t  c 

4th.  Suppose  the  denominator  7^(jx)»  then -Yr7yY=1000c. 

While  the  numerator  remains  the  same,  we  see,  that  as  the  de- 
nominator decreases,  the  value  of  the  fraction  increases.  Hence, 
if  the  denominator  be  less  than  any  assignable  quantity,  that  is  0, 
the  value  of  the  fraction  will  be  greater  than  any  assignable  quan- 
tity, that  is,  infinitely  great.  This  is  designated  by  the  sign  ao, 
that  is  c 

This  is  interpreted  by  saying,  that  no  finite  value  of  x  will 
satisfy  the  equation;  that  is,  there  is  no  number,  which  being 
multiplied  by  0,  will  give  a  product  equal  to  c. 

IV.  When  c  is  0,  and  b  is  either  greater  or  less  than  a. 

If  we  put  a — b  equal  to  d,  then  a:=-=0,  since  c?XO=0;    that 

is,  when  the  product  is  zero,  one  of  the  factors  must  be  zero. 

V.  When  6=a,  and  c=0. 

c       0 
In  this  case,  we  have  x=       ,  =.^,  or  a;XO=;0. 
a — 0    U 

Since  any  quantity  multiplied  by  0,  gives  a  product  equal  to  0, 
any  Jinite  value  of  x  whatever,  will  satisfy  this  equation;  hence,  a; 
is  indeterminate.  On  this  account,  we  say  that  q  is  the  symbol 
of  indetermination  ;  that  is,  the  quantity  which  it  represents,  has 
no  particular  value. 

Review. — 173.  When  is  x  of  the  form  -|-A?  When  is  x  of  the  form 
— A  ?  When  is  x  of  the  form  4,  or  oo  ?  Show  how  the  value  of  a  fraction 
increases,  as  its  denominator  decreases.  What  is  the  value  of  a  fraction 
whose  denominator  is  zero  ?     Of  x  when  c  is  0,  and  h  greater  or  less  than  a  ? 


PROBLEM  OF  THE  COURIERS.         163 

The  form  §  sometimes  arises  from  a  particular  supposition,  when 
the  terms   of  a    fraction  contain    a    common  factor.      Thus,  if 

x~ :-,  and  we  make  6=a,  it  reduces  to -=0  ;    but,  if  we 

a— 6  a— a      " 

cancel  the  common  factor,  a — 6,  and  then  make  6=a,  we  have 

a;=2a.     This  shows,  that  before  deciding  the  value  of  the  unknown 

quantity  to  be  indeterminate,  we  must  see  that  this  apparent  inde- 

termination  has  not  arisen  from  the  existence  of  a  factor,  which, 

bj  a  particular  supposition,  becomes  equal  to  zero. 

The  discussion  of  the  following  problem,  which  was  originally 

proposed  by  Clairaut,  will  serve  to  illustrate  further  the  preceding 

principles,  and  show,  that  the  results  of  every  correct  solution, 

correspond  to  the  circumstances  of  the  problem. 

PROBLEM     OF     THE     COURIERS. 

Two  couriers  depart  at  the  same  time,  from  two  places,  A  and 
B,  distant  a  miles  from  each  other  ;  the  former  travels  m  miles  an 
hour,  and  the  latter,  w  miles;  where  will  they  meet? 

There  are  two  cases  of  this  question. 

I.  When  the  couriers  travel  toward  each  other. 

Let  P  be  the    point  where  they  meet,       A      ^mmmummmmmmmmmmd^     B 

and  a=AB,  the  distance  between  the  ^ 

two  places. 

Let  a;=AP,  the  distance  which  the  first  travels. 

Then  a — a:=BP,  the  distance  which  the  second  travels. 

Then,  the  distance  each  travels,  divided  by  the  number  of  miles 
traveled  in  an  hour,  will  give  the  number  of  hours  he  was  traveling. 

Therefore,  — =  the  number  of  hours  the  first  travels. 
m 

d y^ 

And  =  the  number  of  hours  the  second  travels. 

11 

But  they  both  travel  the  same  number  of  hours,  therefore 

X a — X 

m       n 

nx=am — mx 

a7n 

an 
a — x= — —  . 
m-f-u 

1st.  Suppose  m=?i,  then  a;^^-=^-^,  and  a — 2;=^^;   that  is,  if 

the  couriers  travel  at  the  same  rate,  each  travels  precisely  half 
the  distance. 


164  RAY'S   ALGEBRA,    PART   FIRST. 

2d.  Suppose  ?i=0,  then  x= —  =« ;  that  is,  if  the  second  courier 
m 

remains  at  rest,  the  first  travels  the  whole  distance  from  A  to  B. 
Both  these  results  are  evidently  true,  and  correspond  to  the  cir- 
cumstances of  the  problem. 

II.  When  the  couriers  travel  in  the  same  direction. 

As  before,  let  P  be  the  point  of       A  j  ihm— — ^  P 

meeting,  each  traveling  in  that  direc-  B 

tion,  and  let  a-=AB  the  distance  between  the  places. 
a:=AP  the  distance  the  first  travels. 
X — a=BP  the  distance  the  second  travels. 
Then,  reasoning  as  in  the  first  case,  vre  have 
X  __x — a 
m        n 
nx=mx — am 

am         .  an 

x= ,  and  X — a= . 

m — n  m — n 

1st.  If  we  suppose  m  greater  than  n,  the  value  of  x  will  be  pos- 
itive ;  that  is,  the  couriers  will  meet  on  the  right  of  B.  This  evi- 
dently corresponds  to  the  circumstances  of  the  problem. 

2d.  If  we  suppose  n  greater  than  on,  the  value  of  x,  and  also 
that  of  X — a,  will  be  negative.  This  negative  value  of  x  shows 
that  there  is  some  inconsistency  in  the  question  (Art.  172).  In- 
deed, when  7n  is  less  than  n,  it  is  evident  that  the  couriers  can  not 
meet,  since  the  forward  courier  is  traveling  faster  than  the  hind- 
most. Let  us  now  inquire  how  the  question  may  be  modified,  so 
that  the  value  obtained  for  x  shall  be  consistent. 

If  we  suppose  the  direction  changed  in  which  the  couriers 
travel ;  that  is,  that  the  first  travels  P'  ji^i-Mw-^Miiiiifl^MiiiMiil  B 
from  A,  and  the  second  from  B  to-  A 

ward  F;  and  that  a=AB 
x=AV 
a~{-x=BV,  we  have,  reasoning  as  before, 

X a-\-x 

m        n 

am  ,      ,  an 

x^= ,  and  a+a;= . 

n — m  n — m 

The  distances  traveled  are  now  both  positive,  and  the  question 
will  be  consistent,  if  we  regard  the  couriers,  instead  of  traveling 
toward  P,  as  traveling  in  the  opposite  direction  toward  P'.  The 
change  of  sign,  thus  indicating  a  change  of  direction  (Art.  64). 

3d.  If  we  suppose  m  equal  to  n. 

J     .,.  .  ,  ,     am        -  an 

In  this  case  x  is  equal  to  -^-,  and  x—a=-7r. 


IMPOSSIBLE    PROBLEMS.  165 

As  has  been  already  shown  (Art.  173),  when  the  unknown 
quantity  takes  this  form,  it  is  not  satisfied  by  any  finite  value  ;  or, 
it  is  infinitely  great.  This  evidently  corresponds  to  the  circum- 
stances of  the  problem  ;  for,  if  the  couriers  travel  at  the  same 
rate,  the  one  can  never  overtake  the  other.  This  is  sometimes 
otherwise  expressed,  by  saying,  they  only  meet  at  an  infinite  dis- 
tance from  the  point  of  starting. 

4th.  If  we  suppose  a=0,  then  ic= ,  and  x — a= . 

m — n  m — n 

"When  the  unknown  quantity  takes  this  form,  it  has  been  shown 
already,  that  its  value  is  0.  This  corresponds  to  the  circumstances 
of  the  problem ;  for,  if  the  couriers  are  no  distance  apart,  they 
will  have  to  travel  no  (0)  distance  to  be  together. 

5th.  If  we  suppose  m=^n,  and  a^^O. 

In  this  case,  x=^,  and  x — a=o-  When  the  unknown  quantity 
takes  this  form,  it  has  been  shown  (Art.  173),  that  it  may  have 
any  finite  t-aZwe  whatever.  This,  also,  evidently  corresponds  to  the 
circumstances  of  the  problem ;  for,  if  the  couriers  are  no  distance 
apart,  and  travel  at  the  same  rate,  they  will  be  alicays  together ; 
that  is,  at  any  distance  whatever  from  the  point  of  starting. 

Lastly,  if  we  suppose  n=0,  then  a:= =a ;    that  is,  the  first 

courier  travels  from  A  to  B,  overtaking  the  second  at  B. 

If  we  suppose  n=-^,  then  a;= — — =2a,  and  the  first  travels 
^^  2  m 

twice  the  distance  from  A  to  B,  before  overtaking   the  second. 

Both  results  evidently  correspond   to  the   circumstances  of  the 

problem. 

CASES  OF   INDETERMINATIOM  IN    EQUATIONS   OP   THE   FIRST 
DEGREE,  AND    IMPOSSIBLE    PROBLEMS. 

Art.  174.— An  equation  is  termed  independent, vfhew  the  relation 
of  the  quantities  which  it  contains,  can  not  be  obtained  directly 
from  others  with  which  it  is  compared.     Thus,  the  equation 
x+2y=ll 
2x+5//=26 
are  independent  of  each  other,  since  the  one  can  not  be  obtained 
from  the  other  in  a  direct  manner. 

R  E  v  I  E  w. — 173.  What  is  the  value  of  x  when  h=a  and  c=0  ?  What  is 
the  value  of  a  fraction  whose  terms  arc  both  zero  ?  Show,  that  this  form 
sometimes  arises  from  the  existence  of  a  common  factor,  which,  by  a  par- 
ticular hypothesis,  reduces  to  zero.  Discuss  the  problem  of  the  "  Couriers," 
and  show,  that  in  every  hypothesis  the  solution  corresponds  to  the  cireum- 
st.ancos  of  the  problem. 


166  RAY'S   ALGEBRA,   PART  FIRST. 

The  equations,    x+2y=ll 

2x-|-4y=22,  are  not  independent  of  each  other, 
the  second  being  derived  directly  from  the  first,  by  multiplying 
both  sides  by  2. 

Art.  1 75. — An  equation  is  said  to  be  indeterminate,  when  it  can 
be  verified  by  difierent  values  of  the  same  unknown  quantity. 
Thus,  in  the  equation  x — y=5,  by  transposing  y,  we  have  x=5+y. 

If  we  make  y=\,  a;=6.  If  we  make  2/=2,  a:=7,  and  so  on  ; 
from  which  it  is  evident,  that  an  unlimited  number  of  values  may 
be  given  to  x  and  y,  that  will  verify  the  equation. 

If  we  have  two  equations  containing  three  unknown  quantities, 
we  may  eliminate  one  of  them ;  this  will  leave  a  single  equation, 
containing  two  unknown  quantities,  which,  as  in  the  preceding 
example,  will  be  indeterminate. 

Thus,  if  we  have  x4-3y+2=10 

and  x-f-2y — z=^  6,  if  we  eliminate  x  we  have 
y+22=  4,  from  which  y=^4 — 25!. 

If  we  make  z=\,  y=2,  and  a::=10— 3?/ — 2:=3. 

If  we  make  z^=^lh,  l/=l,  and  a;=52. 

In  the  same  manner,  an  unlimited  number  of  values  of  the  three 
unknown  quantities  may  be  found,  that  will  verify  both  equations. 
Other  examples  might  be  given,  but  these  are  sufficient  to  show, 
that  when  the  number  of  unknown  quantities  exceeds  the  number  of 
independent  equations,  the  problem  is  indeterminate. 

A  question  is  sometimes  indeterminate  that  involves  only  one 
unknown  quantity ;  the  equation  deduced  from  the  conditions,  being 
of  that  class  denominated  identical.     The  following  is  an  example. 

What  number  is  that,  of  which  the  |,  diminished  by  the  f,  is 
equal  to  the  o'g  increased  by  the  3*0? 

Let  a;=  the  number. 

™,        'Sx     2x XX 

Clearing  of  fractions,  45a:— 40a:==3x+2a: 

or,  bx=^bx,  which  will  be  verified  by 
any  value  of  x  whatever. 

Art.  176. — The  reverse  of  the  preceding  case  requires  to  be 
considered;  that  is,  when  the  number  of  equations  is  greater  than 
the  number  of  unknown  quantities.     Thus,  we  may  have 
a:+  y=\0     (1.) 
x-  y=  4     (2.) 
2x— 3y=  5     (3.) 
Each  of  these  equations  being  independent  of  the  other  two, 
one  of  them  is  unnecessary,  since  the  values  of  x  and  y,  which  are 
7  and  3,  may  be  determined  from  any  two  of  them.     When  .'i 


IMPOSSIBLE   PROBLEMS.  167 

problem  contains  more  conditions  than  are  necessary  for  deter- 
mining the  values  of  the  unknown  quantities,  those  that  are  unne- 
cessary, are  termed  redundant  conditions. 

The  number  of  equations  may  exceed  the  number  of  unknown 
quantities,  so  that  the  values  of  the  unknown  quantities  shall  be 
incompatible  with  each  other.     Thus,  if  we  have 
x+  y=.  9     (1.) 
a:+2y=13     (2.) 
2x+3y=21     (3.) 

The  values  of  x  and  y,  found  from  equations  (1)  and  (2),  are 
a:=5,  ?/=4  ;  from  equations  (1)  and  (3),  are  a;=6,  y^=^  ;  and  from 
equations  (2)  and  (3),  are  a;=3,  ?/=5.  From  this  it  is  manifest, 
that  only  two  of  these  equations  can  be  true  at  the  same  time. 

A  question  that  contains  only  one  unknown  quantity,  is  some- 
times impossible.     The  following  is  an  example. 

What  number  is  that,  of  which  the  g  and  \  diminished  by  4,  is 
equal  to  the  |  increased  by  8? 

Let  a;=  the  number,  then  o+q  — 4=-^ +8. 

Clearing  of  fractions,  3x+2a;— 24=5x+48. 
by  subtracting  equals   from  each  side,  0=72 ;  which  shows,  that 
the  question  is  absurd. 

Remark. — Problems  from  which  contradictory  equations  are  deduced, 
nre  termed  irrational  or  impossible.  The  pupil  should  be  able  to  detect  the 
character  of  such  questions  when  they  occur,  in  order  that  his  efforts  may 
not  be  wasted,  in  an  attempt  to  perform  an  impossibility.  A  careful  study 
of  the  preceding  principles,  will  enable  him  to  do  this,  so  far  as  equations 
of  the  first  degree  are  concerned. 

Art.  177. — Take  the  equation  ax — cx=6 — d,  in  which  a  repre- 
Fents  the  sum  of  the  positive,  and  — c  the  sum  of  the  negative 
coefficients  of  a; ;  6  the  sum  of  the  positive,  and  — d  the  sum  of 
the  negative  known  quantities.  This  will  evidently  express  a 
simple  equation  involving  one  unknown  quantity,  in  its  most 
general  form. 

This  gives  [a — c)x=6 — d. 

71 

Let  a — c=m,  and  b — c?=h,  we  then  have  mx=n,  or  a:=: — . 

m 

Now,  since  n  divided  by  m  can  give  but  one  quotient,  we  infer 
that  an  equation  of  the  Jirst  degree  has  but  one  root;  that  is,  in  a 
simple  equation  involving  but  one  unknown  quantity,  there  is  but 
one  value  that  will  verify  the  equation. 

Review. — 174.  When  is  an  equation  termed  independent?  Give  an 
example.  175.  When  is  an  equation  said  to  be  indeterminate?  Give  an 
example.     176.  What  are  redundant  conditions  ? 


168  RAY'S   ALGEBRA,    PART    FIRST. 


CHAPTER  VI. 

FORMATION    OF    POWERS- 
EXTRACTION    OF    THE    SQUARE    ROOT  —  RADICALS    OF    THE 
SECOND    DEGREE. 

IIVVOLUTIOIV,    OR    FORMATION    OF    POWERS. 

Art.  178. — The  term  powej-  is  used  to  denote  the  product  aris- 
ing from  multiplying  a  quantity  by  itself,  a  certain  number  of 
times;  and  the  quantity  which  is  multiplied  by  itself,  is  called  the 
root  of  the  power. 

Thus  a^  is  called  the  second  power  of  a,  because  a  is  taken  twice 
as  a  factor ;  and  a  is  called  the  second  root  of  a\ 

So,  also,  a^  is  called  the  third  power  of  a,  because  aX«X<^==-o^ 
the  quantity  a  being  taken  three  times  as  a  factor ;  and  a  is  called 
the  third  root  of  a'. 

The  second  power  is  generally  called  the  square,  and  the  second 
root,  the  square  root.  In  like  manner,  the  third  power  is  called 
the  cube,  and  the  third  root,  the  cube  root. 

The  figure  indicating  the  power  to  which  the  quantity  is  to  be 
raised,  is  called  the  index,  or  exponent;  it  is  to  be  written  on  the 
right,  and  a  little  higher  than  the  quantity.  (See  Articles  33 
and  35.) 

R  E  M  A  R  K. — A  power  may  be  otherwise  defined  thus :  The  nth  power  of 
a  quantity,  in  the  jirodact  of  n  factors,  each  equal  to  the  quantity;  where  n 
maybe  any  number,  as  2,  3,  4,  and  so  on.  Therefore,  toe  may  obtain  any 
power  of  a  quantity  by  taking  it  as  a  factor  as  many  times  as  there  are  units 
in  the  exponent  of  the poiwr  to  which  it  is  to  be  raised.  This  rule  alone,  is 
sufficient  for  every  question  in  the  formation  of  powers  ;  but,  for  the  more 
easy  comprehension  of  pupils,  it  is  generally  presented  in  detail,  as  in  the 
following  cases. 

CASE    I. 

TO    RAISE    A    MONOMIAL    TO    ANY    GIVEN    POWER. 

Art.  I'y9. — 1.  Let  it  be  required  to  raise  2ab'^  to  the  third 
power. 

According  to  the  definition,  the  third  power  of  2a¥,  will  be  the 
product  arising  from  taking  it  three  times  as  a  factor.     Thus, 
{2a¥Y=2ab''X2ab''X'2ab^=2X2X2aaab^bW 

=2''Xa}+^+^Xb''+'+'=2'Xa'X'Xb''X^=Sa^b^. 
In  this  example,  we  see,  that  the  coefficient  of  the  power  is  found 


FORMATION   OF   POWERS.  169 

by  raising  the  coefficient,  2,  of  the  root,  to  the  given  power;  and, 
that  the  exponent  of  each  letter  is  obtained,  by  multiplying  the 
exponent  of  the  letter  in  the  root,  by  3,  the  index  of  the  required 
power. 

Art.  1§0. — "With  regard  to  the  signs  of  the  different  powers, 
there  are  two  cases. 

First,  when  the  root  is  positive;  and  second,  when  the  root  is 
negative. 

1st.  When  the  root  is  positive.  Since  the  product  of  any  num- 
ber of  positive  factors  is  always  positive,  it  is  evident,  that  if  the 
root  is  positive,  all  the  powers  will  be  positive. 

Thus,  +aX+«=+a'-' 

+aX+«X+«=+«^  and  so  on. 

2d.  When  the  root  is  negative.  Let  us  examine  the  different 
powers  of  a  negative  quantity,  as  — a. 

— «=  first  power,  negative. 

— aX^ — a=-[-a2__  second  -^ovs^qy,  positive. 

— aX — «X — «= — «^=  third  power,  negative. 

— aX — «X — «X — «=+«*=  fourth  power,  positive. 

— aX — «X — «X — «X — «= — ^^=  fifth  power,  negative. 

From  this,  we  see,  that  the  product  of  an  even  number  of  nega- 
tive factors  is  positive,  and  that  the  product  of  an  odd  number  of 
negative  factors  is  negative.  Therefore,  the  even  powers  of  a  neg- 
ative quantity  are  all  positive,  and  the  odd  powers  are  all  negative. 
Hence  we  have  the  following 

RULE, 

FOR    RAISING    A    xMONOMIAL    TO   ANY    GIVEN    POWER. 

Raise  the  numeral  coefficient  to  the  required  power,  and  multiply 
the  exponent  of  each  of  the  letters,  hy  the  exponent  of  the  power.  If 
the  monomial  is  positive,  all  the  powers  will  he  positive ;  hut,  if  it  is 
negative,  all  the  even  powers  will  he  positive,  and  all  the  odd  powers 
negative. 

EXAMPLES. 

1.  Find  the  square  of  Sax^y^ Ans.  9a^x*y\ 

2.  Find  the  square  of  56V Ans.  256*c«. 

3.  Find  the  cube  of  2x'y^ Ans.  8xy. 

4.  Find  the  square  of  — ah'^c Ans.  a^h*c^. 

5.  Find  the  cube  of  — abc^ Ans.  — a^b^c^ 

Review. — 177.  Show,  that  in  an  equation  of  the  first  degree,  the  un- 
known quantity  can  have  but  one  value.  178.  What  does  the  term  power 
denote?  The  term  root?  What  is  the  second  power  of  a?  Why?  The 
third  power  of  a '!  Why  ?  What  is  the  second  power  generally  called  ?  The 
second  root  ?  What  is  the  index  or  exponent  ?  Where  should  it  be  written  ? 
15 


170  RAY'S  ALGEBRA,    PART    FIRST. 

6.  Find  the  fourth  power  of  3a6V Ans.  Sla'b^^cK 

7.  Find  the  fourth  power  of— 3a6V Ans.  81  a*b^h\ 

8.  Find  the  fifth  power  of  abh'd' Ans.  a^b^^c^d}^. 

9.  Find  the  fifth  power  of  —aWcd\     .    .    .     Ans.  —a^b^hH^"^. 

10.  Find  the  sixth  power  of  o?bcH Ans.  o}'^b^c^H^. 

11.  Find  the  seventh  power  of  — Dt^ii^ Ans.  —m^^n^^. 

12.  Find  the  eighth  power  of  ~mn^ Ans.  mhi^^. 

13.  Find  the  cube  of  -3a* Ans.  — 27ai^ 

14.  Find  the  cube  of  —Sxi/ Ans.  —2l3?i/. 

15.  Find  the  fourth  power  of  5a^a?^ Ans.  625a^a;^'. 

16.  Find  the  cube  of  — 4a^a: Ans.  — 64a^a;^. 

17.  Find  the  cube  of  —8xhf Ans.  — 512xy. 

18.  Find  the  seventh  power  of  — 2xyz^.    .    .    Ans.  — 128x'V2;^*. 

19.  Find  the  fourth  power  of  la'a^ Ans.  240 laV^. 

20.  Find  the  fifth  power  of—Sa'xi/zK  .     Ans.  —24Sa^''xy^h'\ 

Art.  181,  case  ii. 

raise  a  polynomial  "  )  any  power. 

RULE. 

Find  the  product  of  the  quantity,  taken  as  a  factor  as  many  times 
as  there  are  units  in  the  exponent  of  the  power. 

Note.  —  This  rule,  and  that  in   the   succeeding  article,  follow  directly 
from  the  definition  of  a  power. 

EXAMPLES. 

1.  Find  the  square  of  ax-^-cy. 

{ax-\-cy)  (ax+cy)=aV+2acic?/+cY^     Ans. 

2.  Find  the  square  of  1 — x Ans.  1 — 2x-\-x'^. 

3.  Find  the  square  of  ic+1 Ans.  a;^-j-2a;+l. 

4.  Find  the  square  of  ax — cy Ans.  aV — 2acxy-\-c^y'^. 

5.  Find  the  square  of  2x^—Sy\    .    .    .  Ans.  4x*— 12xy+9y*. 

6.  Find  the  cube  of  a-\-x Ans.  <x'+3(Z^a:+3ax''^+a:'. 

7.  Find  the  cube  of  x — y Ans.  ic^ — 3x'^y-\'3xy^ — y^. 

8.  Find  the  cube  of  2a;— 1 Ans.  Sx^— 12x2+6a;— 1. 

9.  Find  the  fourth  power  of  c — x. 

Ans.  c* — 4c^x-{-6c'^x'^ — 4ca^^x*. 

10.  Find  the  square  of  a+6+c. 

Ans.  a^-\~2ab-{-b^+2ac+2bc+c'. 

11.  Find  the  square  of  a — b-\-c — d. 

Ans.a''—2ab-{-b^+2ac—2ad+c^—2bc-\-2bd—2cd+d:'. 

12.  Find  the  cube  of  2a;2— 3a;+l. 

Ans.  8x«— 36x^+66a;*— 63a:'+33x2— 9;p-f  1 . 


FORMATION   OF   POWERS.  171 

Akt.  182.  CASK  III. 

TO    RAISE    A    FRACTION    TO    ANY    POWER. 
RULE. 
Raise  both  numerator  atid  denominator  to  the  required  power  brj 
actual  multiplication, 

EXAMPLES. 

1 .  Find  the  square  of ^. 

a-\'b     a+b_a''+2ab+¥ 
c—dT  c—d~~  c'—2cd'-{-d'' 

2.  Find  the  square  of  h- Ans.  rr-y. 

3.  Find  the  cube  of r- Ans.  ^  — --. 

2y?  4:X* 

4.  Find  the  square  of — ^z-- Ans.  ;^— . 

3y  9/ 

5.  Find  the  cube  of  -^ Ans.  ~^^K- 

52''  1253^ 

6.  Find  the  square  of  '—-7^ Ans.    .,~^^       -. 

ic+3  a;^+6a;+9 

7.  Find  the  cube  of  ^t~'^>.     .  Ans.  8-»V-3.yW^) 

Q    t;,.    ,  .,  „  2{m—n)  4{m^—2mti-\-n'^) 

8.  Find  the  square  of  07— T~-     •    •    •  ^^«-  «w    2  ,  o T"^  • 

3{m-{-n)  y{m^-j-2mn-j-n^ 

BINOMIAL    THEOREM. 

Art.  183. — The  Binomial  Theorem  (discovered  by  Sir  Isaac 
Newton),  explains  the  method  of  raising  the  sum  or  difference  of 
any  two  quantities  to  any  given  power,  by  means  of  certain  rela- 
tions, that  are  always  found  to  exist  between  the  exponent  of  the 
power  and  the  different  parts  of  the  required  result. 

To  discover  what  these  relations  are,  we  shall  first,  by  means  of 
multiplication,  find  the  different  powers  of  a  binomial,  when  both 
terms  are  positive ;  and  next,  when  one  term  is  positive,  and  the 
other  negative. 

R  E  V I  E  w. — 179.  In  raising  2ab^  to  the  third  power,  how  is  the  coSflBcient 
of  the  power  found  ?  How  is  the  exponent  of  each  letter  found  ?  180.  When 
the  root  is  positive,  what  is  the  sign  of  the  different  powers  ?  When  it  ia 
negative?     What  is  the  rule  for  raising  a  monomial  to  any  given  power? 

181.  What  is    the    rule   for  raising   a   polynomial   to  any  given  power? 

182.  What  is  the  rule  for  raising  a  fraction  to  any  power?     183.  What  does 
the  Binomial  Theorem  explain  ? 


172  RAY'S   ALGEBRA,    PART   FIRST. 

1.  We  will  first  raise  a+6  to  the  fifth  power. 

a-\-  b 

a-\-  b 

a^+  ab 

+    ct6+     b^' 

a^-{-2ab-\-  W= second  power  of  a+6,  or  (a4-^)'- 

aH-  & 

a'+2a^6+     a  b' 

a^6+  2ab''-[-  b^ 

a^-\r^d^b-\-  3«  b''-\-   b^=   ....   third  power  of  a+6,  or  [a-^bf. 

a+b 

a*-\-3a^b+  3d'b'+     ab^ 

-h  a^6+  3a^6M-  3a  6^+&* 

a*+4a-^6+   Qd'b'-^-  Aab^+b'= [a^bf. 

«+6 . 

a5+4a*64-   Qd'b'^  4.tW-\-   ab* 

+  a'b+  4a-V/^+   6a'b^+4ab*+b' 

a^-{-5a*b+l0a^b'+l0d'b^-\-5ab*+b^= (a+6)». 

The  first  letter,  as  a,  is  called  the  leading  quantity ;  and  the 
second  letter,  as  b,  ih.Q  following  quantity. 
We  will  next  raise  a — b  to  the  fifth  power. 

a  —  b 
a  —   b 

a^ —  ab 

—  ab+     y 

a^— 2a6+     b''= .  [a—by. 

a  —  b 

a^—2a^b-\-     a/b" 

—  a'^b-^  2a  6^—  b^         

a-"'— 3a264-  3a  5^—  6^^ " {a—bf. 

a —  b 

a*—3a^b+  Sd'b''—     ab^ 

—  a^b+  SaW—  SaP+  b* 

a*—4a^b+  Gd'b'—  4a  b^+  ¥=     .........  ^     {a—b)\ 

a  —  b 

a5_4a*6+  Qa^b'—  4d'W-\-  ab*  ' 

—  a*b+  4a%''—  QaVj^J^4ab*-  b^ 

a^—5a*b  -f- 1  Oa'b^— 1 0a''b^-\-5ab*-b^= ~     {a—b)\ 


FORMATION   OF    POWERS.  173 

Art.  1§4. — In  examining  the  different  parts  of  which  these 
results  consist,  there  are  evidently  four  things  to  be  considered. 

1st.  The  number  of  terms  of  the  power. 

2d.    The  signs  of  the  terms. 

3d.    The  exponents  of  the  letters. 

4th.  The  coefficients  of  the  terms. 

We  shall  examine  these  separately. 

1st.  Of  the  number  of  terms. 

By  examining  either  of  these  examples,  we  see,  that  the  second 
power  has  three  terms,  the  third  power  has  four  terms,  the  fourth 
power  has  fve  terms,  the  ffth  power  has  six  terms ;  hence,  we 
infer,  that  the  number  of  terms  in  any  power  of  a  binomial,  is  one 
greater  than  the  exponent  of  the  power. 

2d.  Of  the  signs  of  the  terms. 

From  an  examination  of  the  examples,  it  is  evident,  that  tvhen 
both  terms  of  the  binomial  are  positive,  all  the  terms  will  be  positive. 
When  the  first  term  is  positive,  and  the  second  negative,  all  the  odd 
terms  will  be  positive,  and  the  even  terms  negative. 

Note  . — By  the  odd  terms  are  meant  the  1st,  3d,  5th,  and  so  on ;  and, 
by  the  even  tei-ms,  the  2d,  4th,  6th,  and  so  on. 

3d.  Of  the  exponents  of  the  letters. 

If  we  omit  the  coefficients,  the  remaining  parts  of  the  fifth 
powers  of  a-\-b  and  a — b,  are 

[a^-bf a5+a*6+a='62+a2Z>3+aM+65. 

[a—bf a^—a%+a%''—d'b^^a¥—bK 

An  examination  of  these  and  the  other  different  powers  of  a-\-h 
and  a — b,  shows,  that  the  ejjponents  of  the  letters  are  governed  by 
the  following  laws : 

1st.  The  exponent  of  the  leading  letter  in  the  first  term,  is  the  same 
as  that  of  the  power  of  the  binomial;  and  the  exponents  of  this  letter 
in  the  other  terms,  decrease  by  unity  from  left  to  right,  until  the  last 
term,,  which  does  not  contain  the  leading  letter, 

2d.  The  exponent  of  the  second  letter  in  the  second  term  is  one; 
and  the  other  exponents  of  this  letter  increase,  by  unity,  from  left  to 
right,  until  the  lad  term,  in  which  the  exponent  is  the  same  as  that 
of  the  power  of  the  binomial. 

3d.  Tlic  Slim  of  the  exponents  of  the  two  letters  in  any  term  is 
always  the  same,  and  is  equal  to  the  power  of  the  binomial. 

R  E  V  I  E  w. —  184.  In  examining  the  different  powers  of  a  binomial,  what 
four  things  are  to  be  considered?  What  is  the  number  of  terms  in  any 
power  of  a  binomial  ?  Give  examples.  When  both  terms  of  a  binomial  are 
positive,  what  are  the  signs  of  the  terms  ?  When  one  term  is  positive,  and 
the  other  negative,  what  are  the  signs  of  the  odd  terms  ?  Of  the  even 
terms?     What  is  the  exponent  of  the  leading  letter  in  the  first  term? 


174  RAY'S   ALGEBRA,    PART   FIRST. 

The  pupil  may  now  employ  these  principles,  in  writing  the  dif- 
ferent powers  of  binomials  without  the  coefficients,  as  in  the  fol- 
lowing examples. 

{x+ijY  .    .    .  x^-{-x'^i/+xi/^-\-i/. 

{x—ijY  .    .    .  x^—:x?y+xhf—xy^'\-yK 

[x+ijf  .    .    .  x^-{-x^y+3?if+xh/+xy^-^7f. 

[x—yY  .    .    .  x^ — a^y-\-x*y^ — x^y^-\-xh/ — xtf+i^. 

[x — yY  .    .    .  x' — x^y-\-x''y^ — x^y^-\-3(^y'^ — x^if'-\-xy^ — y"^. 

{x-\-yY  .    .    .  3i?-\-x'^y-\-x^y^-\-x^y^-^x*}/-^x^y^-\-x^y^-\-xy^-\-'if. 

Of  the  coefficients. 

An  inspection  of  the  different  powers  of  (a+6)  and  (a — h), 
plainly  shows, 

That  the  coefficient  of  the  Jirst  term  is  always  1;  and  the  coeffi- 
cient of  the  second  term  is  the  same  as  that  of  the  power  of  the  binomial. 

The  law  of  the  succeeding  coefficients  is  not  so  readily  seen ;  it 
is,  however,  as  follows : 

If  the  coefficient  of  any  term  be  midtiplied  by  the  exponent  of  the 
leading  letter,  and  the  product  be  divided  by  the  number  of  that  term 
from  the  left,  the  quotient  will  be  the  coefficient  of  the  next  term. 

Omitting  the  coefficients,  the  terms  of  a+6  raised  to  the  sixth 
power,  are         a'^-\-a^b+an/^a^b^+a%^+ab^+b^. 

The  coefficients,  according  to  the  above  principles,  are 
,    P  ^5    15X4    20X3    15X2    6X1 
1,  t>,     2    ,       j^    .       4    '       5    '      6   * 

or,  1,  6,     15,        20,         15,  6,  1. 

Hence,  {a-^bY-=-a^^{Sa''b-\-\ba*b''+20a^b^+\^a''b^-\-Qab^-\-b\ 
From  this,  we  see,  that  the  coefficients  of  the  following  terms 
are  equal :  the  first  and  the  last ;  the  second  from  the  first,  and  the 
second  from  the  last;  the  third  from  the  first  and  the  third  from 
the  last,  and  so  on.  Hence,  it  is  only  necessary  to  find  the  coeffi- 
cients of  half  the  terms,  when  their  number  is  even,  or  one  more 
than  half,  when  their  number  is  odd ;  the  remaining  coefficients 
being  equal  to  those  already  found. 

EXAMPLES. 

1.  Raise  x-\-y  to  the  third  power.  Ans.  x^^^xhj-\-^xy'^-\-y^. 

2.  Raise  [x — y)  to  the  fourth  power. 

Ans.  X-*— 4x^?/4-6a;^?/^ — 4txi/-\'y^. 

3.  Raise  M-\-n  to  the  fifth  power. 

Ans.  m^+57/t*«+10/;i'yt'^+10»i^w' -  5mw*-|-7<\ 

R  E  V I  li  w.— ]  84.  How  do  the  exponents  of  the  leading  letter  decrease 
from  left  to  right?  What  is  the  exponent  of  the  second  letter  in  the  tirst 
term  ?  In  the  second  term  ?  How  do  the  exponents  of  the  second  letter 
increase  from  left  to  right  ?     To  what  is  the  coefficient  of  the  first  term  equal  ? 


FORMATION  OF  POWERS.  175 

4.  Raise  x — z  to  the  sixth  power. 

Ans.  x'-Gx^z-i- 1 5xV—20xV+  150:^—6x2^+28. 

5.  AVhat  is  the  seventh  power  of  a-\-b? 

6.  What  is  the  eighth  power  of  m — Ji  ?  Ans.  7n^ — Sw'w 

7.  Find  the  ninth  power  of  x — i/.  Ans.  x^ — 9x^i/-{-S6x^y'^ 

— 84a;V+  l2GxY~  1 26xY+S43^i/^-S6x'i/-^-^9x7/—f. 

8.  Find  the  tenth  power  of  a+6. 

Ans.  a^''+I0a96+45a862+120a^63^210a«6*+252a55H-210a*i« 
+ 1 20a^b-'+46a'b^+ 1  Oab^-^-b'". 

Art.  185. — The  Binomial  Theorem  may  be  used  to  find  the 
different  powers  of  a  binomial,  when  one  or  both  terms  consist  of 
two  or  more  quantities. 

1.  Find  the  cube  of  2x  —ac^. 

Let  2x=m,  and  ac^=n  ;  then  2x — ac^=m — n. 
{m — nY=m^ — Sm^n-i-Smn^ — rv^ 
m  =2a:  ii  =a  c^ 

Substituting  these  values  of  the  different  powers  of  m  and  n, 
in  the  equation  above,  and  we  have 

l2x—ac'Y=Sx'—3X4x'Xac'+SX2xXah*—ah^ 
=Sx^—  1 2ac'x'--\-Gah*x—a^c\ 

2.  Find  the  cube  of  2a-3b.       Ans.  Sa^— 36a26+54a6'^-276». 

3.  Find  the  fourth  power  of  m-j-2n. 

Ans.  m*+8m3n+24;«2n2+32wn3-|-16;t*. 

4.  Find  the  third  power  of  4ax^-{-Saj. 

Ans.  64a3x6+ 1 44a'cx*y+ 1  OSac^xy +27c3a/». 

5.  Find  the  fourth  power  of  2x — 5z. 

Ans.  lQx'—lQ0xh+600x'z'-l000xz^-i-62ijz*. 
Art.  186.— TLs  Binomial  Theorem  may  likewise  be  used  to 
raise  a  trinomial  or  quadrinomial  to  any  power,  by  reducing  it  to 
a  binomial  by  substitution,  and  then,  after  this  has  been  raised  to 
the  required  power,  restoring  the  values  of  the  letters. 
1.  Find  the  second  power  of  a-^b-'rc 
Let  b-{-c=x;  then  a-\-b-\-c=a~['X. 

[a+xY=a^+2ax+x^ 
2ax  =2a[b-\-c) 

x'={b^cY=-b'^+2bc^c' 
Then  [a-\-b-\-cf=a^-^2ab^2ac-\-b''+2bc+c\ 

Review. — 184.  Of  tho  second  term?  How  is  the  coefficient  of  any 
other  term  found  ?     Of  what  terms  are  the  coefficients  equal  ? 


176  RAY'S   ALGEBRA,    PART   FIRST. 

2.  Find  the  third  power  of  x-{-y-{-z. 

3.  Find  the  second  power  of  a-\-b-\-c-\-d. 

Ans.  a-'+2ab-\-¥+2ac+2bc+c^-}-2ad-\-2bd+2cd+d\ 


EXTRACTION    OF     THE     SQUARE    ROOT. 

EXTRACTION    OF   THE   SQUARE  ROOT   OF   IVUMBERS. 

Art.  187. — The  second  7'ooi,  or  square  root  of  a  number,  is  that 
number,  which  being  multiplied  by  itself,  will  produce  the  given 
number.     Thus,  2  is  the  square  root  of  4,  because  2X2=4. 

The  process  of  finding  the  second  root  of  a  given  number,  is 
called  the  extraction  of  the  square  root. 

Art.  188. — The  first  ten  numbers  are 

1,  2,  3,     4,     5,     6,     7,     8,     9,     10, 
and  their  squares  are 

1,  4,  9,  IG,  25,  36,  49,  64,  81,  100. 

The  numbers  in  the  first  line,  are  alsj  the  square  roots  of  the 
numbers  in  the  second. 

We  see,  from  this,  that  the  square  root  of  a  number  between  1 
and  4,  is  a  number  between  1  and  2  ;  the  square  root  of  a  num- 
ber between  4  and  9,  is  a  number  between  2  and  3  ;  the  square 
root  of  a  number  between  9  and  16,  is  a  number  between  3  and 
4,  and  so  on. 

Since  the  square  root  of  1  is  1,  and  of  any  number  less  than 
100,  is  either  one  figure,  or  one  figure  and  a  fraction,  therefore, 
when  the  number  of  places  of  figures  in  a  number  is  not  more  than 
TWO,  the  number  of  places  of  figures  in  the  square  root  will  be  one. 

Again,  take  the  numbers 

10,    20,    30,      40,      50,      60,      70       80,      90,      100, 
their  squares  are 
100,  400,  900,  1600,  2500,  3600,  4900,  6400,  8100,  10000. 

From  this  we  see,  that  the  square  root  of  100  is  ton  ;  and  of 
any  number  greater  than  100,  and  less  than  10000,  the  square 
root  will  be  less  than  100;  that  is,  when  the  number  of  places  of 
figures  is  more  than  two,  and  not  more  than  four,  the  number  of 
places  of  figures  in  the  square  root  will  be  two. 

In  the  same  manner,  it  may  be  shown,  that  when  the  number 
of  places  of  figures  in  a  given  number  are  more  than  four,  and 
not  more  than  six,  the  number  of  places  in  the  square  root  will  be 
three,  and  so  on.     Or  thus :  when  the  number  of  places  of  figures 


EXTRACTION  OF  THE  SQUARE  ROOT.  177 

in  the  number  is  either  one  or  two,  there  will  be  one  figure  in  the 
root ;  when  the  number  of  places  is  either  three  or  four,  there  will 
be  two  figures  in  the  root ;  when  the  number  of  places  is  either 
Jive  or  six,  there  will  be  three  figures  in  the  root,  and  so  on. 

Art.  189. — Every  number  may  be  regarded  as  being  composed 
of  tens  and  units.  Thus,  23  consists  of  2  tens  and  3  units ;  256 
consists  of  25  tens  and  6  units.  Therefore,  if  we  represent  the 
tens  by  t,  and  the  units  by  w,  any  number  will  be  represented  by 
t-{-u,  and  its  square,  by  the  square  of  t-\-u,  or  [t-\-uY. 
[t-^uy=f+2tu+u^=f+[2t+u)u. 

Hence,  the  square  of  amj  number  is  composed  of  the  square  of  the 
tens,  j)lus  a  quantity,  consisting  of  twice  the  tens  plus  the  units,  mul- 
tiplied by  the  units. 

Thus,  the  square  of  23,  which  is  equal  to  2  tens  and  3  units,  is 

2  tens  squared  =(20)2=400 
(2  tens  +  3  units)  multiplied  by  3=(40+3)X3^129 

529 

1 .  Let  it  now  be  required  to  extract  the  square  root  of  529. 

Since  the  number  consists  of  three  places  529123 

of  figures,  its  root  will  consist  of  two  places,  400] 

according  to  the  principles  in  Art.  188;  we     20X2=40  129 
therefore  separate  it  into  two  periods,  as  in  3 

the  margin.  43  129 

Since  the  square  of  2  tens  is  400,  and  of  3  tens,  900,  it  is  evi- 
dent, that  the  greatest  square  contained  in  500,  is  the  square  of 
2  tens  (20)  ;  the  square  of  two  tens  (20)  is  400  ;  subtracting  this 
from  529,  the  remainder  is  129. 

Now,  according  to  the  preceding  theorem,  this  number  129  con- 
sists of  twice  the  tens  plus  the  units,  multiplied  by  the  units;  that 
is,  by  the  formula,  it  is  [2t-\-u)u.  Now,  the  product  of  the  tens 
by  the  units  can  not  give  a  product  less  than  tens ;  therefore,  the 
unit's  figure  (9)  forms  no  part  of  the  double  product  of  the  tens 
by  the  units.  Then,  if  we  divide  the  remaining  figures  (12)  ])y 
the  double  of  the  tens,  the  quotient  will  be  the  unit's  figure,  or  a 
figure  greater  than  it. 

Review.  —  187.  What  is  the  square  root  of  a  number?  Give  an  ex- 
ample. 188.  When  a  number  consists  of  only  one  figure,  what  is  the  great- 
est number  of  figures  in  its  square  ?  Give  examples.  When  a  number 
consists  of  two  places  of  figures,  what  is  the  greatest  number  of  figures  in 
its  square?  Give  examples.  What  relation  exists  between  the  number  of 
places  of  figures  in  any  number,  and  the  number  of  places  in  its  square  ? 
189.  Of  what  may  every  number  be  regarded  as  being  composed  ?  Prove 
this,  and  then  illustrate  it. 


178  RAY'S    ALGEBRA,    PART  FIRST. 

We  then  double  the  tens,  which  makes  4  (2/),  and  dividing  this 
into  12,  get  3  [ii]  for  a  quotient;  this  is  the  unit's  figure  of  the 
root.  This  unit's  figure  (3)  is  to  be  added  to  the  double  of  the 
tens  (40),  and  the  sum  multiplied  by  the  unit's  figure.  The  double 
of  the  tens  plus  the  units,  is  40+3=43  {2t-{-ti) ;  multiplying  this 
by  3  («),  the  product  is  129,  which  is  the  double  of  the  tens  plus 
the  units,  multiplied  by  the  units.  As  there  is  nothing  left  after 
subtracting  this  from  the  first  remainder,  we  conclude 
that  23  is  the  exact  square  root  of  529.  529)23 

In  squaring  the  tens,  and  also  in  doubling  them,  it  4        ~~ 

is  customary  to  omit  the  ciphers,  though  they  are  un-     48iT29 
derstood.     Also,the unit's  figure  is  added  to  the  double  jl29 

of  the  tens,  by  merely  writing  it  in  the  unit's  place. 
The  actual  operation  is  usually  performed  as  in  the  margin. 

2.  Let  it  be  required  to  extract  the  square  root  of  55225. 

Since  this  number  consists  of  five  places  of  figures,  its  root  will 
consist  of  three  places,  according  to  the  principles  in  Art.  188  ; 
we  therefore  separate  it  into  three  periods.  5522*^12^5 

In  performing  this  operation,  we  find  the  square       ^  

root  of  the  number  552,  on  the  same  principle  as 

in  the  preceding  example.     We  next  consider  the  _jl^^ 
23  as  so  many  tens,  and  proceed  to  find  the  unit's 


figure  (5)  in  the  same  manner  as  in  the  preceding  465|2325 
example.     Hence  the  23'^o 

RULE, 

FOR    THE    EXTRACTION    OF    THE    SQUARE    ROOT    OF    WHOLE    NUMBERS. 

1st.  Separate  the  given  number  into  periods  of  two  places  each, 
beginning  at  the  unit's  p)lace.  (The  left  period  will  often  contain 
but  one  figure.) 

2d.  Find  the  greatest  square  in  the  left  period,  and  place  its  root 
on  the  right,  after  the  manner  of  a  quotient  in  division.  Subtract 
the  square  of  the  root  from  the  lej't  period,  and  to  the  remainder  bring 
down  the  next  period  for  a  dividend. 

3d.  Double  the  root  already  found,  and  place  it  on  the  left  for  a 
divisor.  Find  hoio  many  times  the  divisor  is  contained  in  the  divi- 
dend, exclusive  of  the  right  hand  figure,  and  place  the  fgure  in  the 
root,  and  also  on  the  right  of  the  divisor. 

4th.  Midliply  the  divisor  thus  increased,  by  the  last  fgure  of  the 
root;  subtract  the  ptroduct  from  the  dividend,  and  to  the  remainder 
bring  down  the  next  period  for  a  new  dividend. 

,  5th.   Double  the  whole  root  already  found,  for  a  new  divisor,  and 
continue  the  operation  as  before,  until  all  the  periods  are  brought  down. 

Review. — 189.  Extract  the  square  root  of  529,  and  show  the  reason  for 
each  step,  by  referring  to  the  formula. 


EXTRACTION  OF  THE  SQUARE  ROOT. 


179 


XoTK. — If,  in  any  case,  the  dividend  will  not  contain  the  divisor,  tho 
right  hand  figure  of  the  former  being  omitted,  place  a  zero  in  the  root,  and 
also  at  the  right  of  the  divisor,  and  bring  down  the  next  period. 

Art.  190. — In  Division,  when  the  remainder  is  greater  than  the 
divisor,  the  last  quotient  figure  may  be  increased  by  at  least  1; 
but  in  extracting  the  square  root,  the  remainder  may  sometimes 
be  greater  than  the  last  divisor,  while  the  last  figure  of  the  root 
can  not  be  increased.  To  know  when  any  figure  may  be  increased, 
the  pupil  must  be  acquainted  with  the  relation  that  exists  between 
the  squares  of  two  consecutive  numbers. 

Let  a  and  a-{- 1  be  two  consecutive  numbers. 

Then  (a+l)^=a^+2a+l,  is  the  square  of  the  greater. 
{aY^=a^  is  the  square  of  the  less. 

Their  difference  is  2a4-l. 

Hence,  the  difference  of  the  squares  of  two  consecutive  numbers,  is 
equal  to  twice  the  less  number,  increased  by  unity.  Consequently, 
when  the  remainder  is  less  than  twice  the  part  of  the  root  already 
found,  plus  unity,  the  last  figure  can  not  be  increased. 

Extract  the  square  root  of  the  following  numbers. 


1.  4225 Ans.  65. 

2.  9409 Ans.  97. 

8.  15129.  .  .  .  Ans.  123. 

4.  12040D.   .  .  Ans.  347. 

5.  289444.   .  .  Ans.  538. 

6.  498436.  .  .  .  Ans.  706. 


7.  678976.  . 

8.  950625.  .  . 

9.  363609.  .  . 

10.  1525225.  .  . 

11.  1209996225. 


Ans.  824. 
Ans.  975. 
Ans.  603. 
Ans.  1235. 
A. 34785. 


12.  412252416.  Ans.  20304. 


that 


EXTRACTION  OF  THE   SQUARE    ROOT    OF  FRACTIOIVS. 

Art.  191. — Since  |x|=-|,  therefore,  the  square  root  of  |  is  |, 

5,     /'*__- --_:^.     Hence,  when  both  terms  of  a  fraction  are 
^'9~T/9~3 

perfect  squares,  its  square  root  will  be  found,  by  extracting  the  square 
root  of  both  terms. 

Before  extracting  the  square  root  of  a  fraction,  it  should  be 
reduced  to  its  lowest  terms,  unless  both  numerator  and  denomina- 
tor are  perfect  squares.  The  reason  for  this,  will  be  seen  by  tho 
following  example. 

Find  the  square  root  of  ^f. 
12    4X3 


Here,  ^ 


27    9X3" 


Now,  neither  12  nor  27  are  perfect  squares: 


R  K  VIEW. — 189.  What  is  the  rule  for  extracting  the  square  root  of  num- 
bers ?  190.  What  is  the  difference  between  the  squares  of  two  consecutive 
numbers  ?     When  may  any  figure  of  the  quotient  be  increased.' 


180  RAY'S   ALGEBRA,    PART   FIRST. 

but,  by  canceling  the  common  factor  3,  the  fraction  becomes  ^,  of 
which  the  square  root  is  | . 

When  both  terms  are  perfect  squares,  and  contain,  a  common 
factor,  the  reduction  may  be  made  either  before,  or  after  the  square 
root  is  extracted.     Thus,  y'l^=^=^;  or,  3^  =  9,  and  |/|=:j. 

Find  the  square  root  of  each  of  the  following  fractions. 

2.  o^A Ans.  r, 


q      '07  1  A„„     3 


5.  tVoVo Ans.  tVo. 

•    J  0000  00-  •    •    •  ^^-n's-  TOOO* 

Art.  193. — A  number  whose  square  root  can  be  exactly  ascer- 
tained, is  termed  a  jjerfecf  square.  Thus,  4,  9,  16,  &c.,  are  per- 
fect squares.     Comparatively,  these  numbers  are  few. 

A  number  whose  square  root  can  not  be  exactly  ascertained,  is  ■ 
termed  an  imperfect  square.  Thus,  2,  3,  5,  6,  &c.,  are  imperfect 
squares. 

Since  the  difference  of  two  consecutive  square  numbers,  a'^  and 
a*'^+2a4-l,  is  2a+l  ;  therefore,  there  are  always  2a  imperfect 
squares  between  them.  Thus,  between  the  square  of  4(16),  and 
the  square  of  5(25),  there  are  8(2a=2X4)  imperfect  squares. 

A  root  which  can  not  be  exactly  expressed,  is  called  a  siird,  or 

irrational  root.     Thus  y2  is  an  irrational  root;  it  is  1.414+. 

The  sign  -f ,  is  sometimes  placed  after  an  approximate  root,  to 
denote  that  it  is  less,  and  the  sign  — ,  that  it  is  greater  than  the 
true  root. 

It  might  be  supposed,  that  when  the  square  root  of  a  whole 
number  can  not  be  expressed  by  a  whole  number,  that  it  might  be 
found  exactly  equal  to  some  fraction.  We  will,  therefore,  show, 
that  the  square  root  of  an  imperfect  square,  can  not  he  a  fraction. 

Let  c  be  an  imperfect  square,  such  as  2,  and  if  possible,  let  its 

square  root  be  equal  to  a  fraction  j,  which  is  supposed  to  be  in  its 

lowest  terms. 

Then  ■Jc=^t  ;  and  ^=-7^,  by  squaring  both  sides. 

Now,  by  supposition,  a  and  b  have  no  common  factor,  therefore, 
their  squares,  a^  and  b"^,  can  have  no  common  factor,  since  to  square 

a^ 
a  number,  we  merely  repeat  its  factors.      Consequently,  jj  must 

be  in  its  lowest  terms,  and  can  not  be  equal  to  a  whole  numljcr. 

a'^ 
Therefore,  the  equation  c=— ,  is  not  true  ;  and  hence,  the  suppo- 
sition is  false  upon  which  it  is  founded;  that  is,  that  i/c=j;  there- 
fore, the  square  root  of  an  imperfect  square  can  not  be  a  fraction. 


EXTRACTION  OF  THE  SQUARE  ROOT.  181 


APPROXIMATE     SQUARE     ROOTS. 

Art.  193. — To  illustrate  the  method  of  finding  the  approximate 
square  root  of  an  imperfect  square,  let  it  be  required  to  find  the 
square  root  of  2  to  within  -3. 

Reducing  2  to  a  fraction  whose  denominator  is  9  (the  square  of 
3,  the  denominator  of  ihe  fraction  -g),  we  have  2=^. 

Now,  the  square  root  of  18  is  greater  than  4,  and  less  than  5 
therefore,  the  square  root  of  ^f  is  greater  than  -3,  and  less  than  | ; 
therefore,  -|  is  the  square  root  of  2  to  within  less  than  |. 

Hence  the 

RULE, 

FOR   EXTRACTING    THE    SQUARE    ROOT    OF    A    WHOLE  NUMBER  TO  WITHIN 
A    GIVEN    FRACTION. 

MuUiply  the  given  number  hy  the  square  of  the  denominator  of  the 
fraction  which  determines  the  degree  of  approximation ;  extract  the 
square  root  of  this  product  to  the  nearest  unit,  and  divide  the  residt 
hy  the  denominator  of  the  fraction. 

EXAMPLES. 

1.  Find  the  square  root  of  5  to  within  ^ Ans.  2x. 

2.  Find  the  square  root  of  7  to  within  j^.  . 

3.  Find  the  square  root  of  15  to  within  t^'^. 

4.  Find  the  square  root  of  27  to  within  3'^. 

5.  Find  the  square  root  of  14  to  within  y  q. 

6.  Find  the  square  root  of  15  to  within  y^-(j. 
Since  the  square  of  10  is  100,  the  square  of  100,  10000,  and  so 

on,  the  number  of  ciphers  in  the  square  of  the  denominator  of  a  dec 
imal  fraction  is  equal  to  twice  the  number  in  the  denominator  itself. 
Therefore,  ivhen  the  fraction  which  determines  the  degree  of  approxi- 
mation is  a  decimal,  it  is  merely  necessary  to  add  two  ciphers  for  each 
decimal  place  required;  and,  after  extracting  the  root,  to  point  off 
from  the  right,  one  place  of  decimals  for  each  tivo  ciphers  added. 

7.  Find  the  square  root  of  2  to  six  places  of  decimals. 

Ans.  1.414213. 

8.  Find  the  square  root  of  5  to  five  places  of  decimals. 

Ans.  2.23606. 

Revikw. — 191.  How  is  the  square  root  of  a  fraction  found,  when  both 
terms  are  perfect  squares  ?  192.  When  is  a  number  a  perfect  square  ? 
Give  examples.  When  is  a  number  an  imperfect  square?  How  can  you 
determine  the  number  of  imperfect  squares  between  any  two  consecutive 
perfect  squares  ?  What  is  a  root  called,  which  can  not  be  exactly  expressed  ? 
Prove  that  the  square  root  of  an  imperfect  square  can  not  he  a  fraction. 
193.  How  do  you  find  the  approximate  square  root  of  an  imperfect  square 
to  within  any  given  fraction  ?  What  is  the  rule,  when  the  fraction  which 
determines  the  degree  of  approximation,  is  a  decimal  ? 


Ans.  2j%. 

Ans.  3|?. 
.  Ans.  5|. 
.  Ans.  3.7. 

Ans.  3.87. 


182  RAY'S   ALGEBRA,    PART    FIRST. 


9.  Find  the  square  root  of  10 Ans.  3.162277-f  •. 

10.  Find  the  square  root  of  101 Ans.  10.049875+. 

11.  Find  the  square  root  of  60 Ans.  7.74596-f . 

Art.  194. — To  find  the  approximate  square  root  of  a  fraction. 

1.  Let  it  be  required  to  find  the  square  root  of  |  to  within  \. 

Now,  since  the  square  root  of  21  is  greater  than  4,  and  less 
than  5,  therefore,  the  square  root  of  4^  is  greater  than  7,  and  less 
than  I ;  hence  |  is  the  square  root  of  f  to  within  less  than  \. 

Hence,  r/"  ice  multiphj  the  numerator  of  a  fraction  hy  its  denomi- 
nator, then  extract  the  square  root  of  the  product  to  the  nearest  unity 
and  divide  the  result  hy  the  denominator,  the  quotient  will  he  the 
square  root  of  the  fraction  to  within  one  of  its  equal  paiis. 

2.  Find  the  square  root  of  j'j  to  within  j j Ans.  /y. 

3.  Find  the  square  root  of  j\  to  within  j- Ans.  |. 

4.  Find  the  square  root  of  j§  to  within  -^3 Ans.  \\- 

Since  any  decimal  may  be  written  in  the  form  of  a  fraction 

having  a  denominator  a  perfect  square,  by  adding  ciphers  to  both 
terms  (thus,  .4=yVo=TVo^0o>  &c.),  therefore,  the  square  root  may 
be  found,  as  in  the  method  of  approximating  to  the  square  root  of 
a  whole  number,  by  annexing  ciphers  to  the  given  decimal,  until  the 
numher  of  decimal  places  shall  he  equal  to  douhlc  the  number  required 
in  the  root.  Then,  after  extracting  the  root,  pointing  of  from  the 
right,  the  required  numher  of  decimal  places. 
Find  the  square  root 

5.  Of  .6  to  six  places  of  decimals Ans.  .774596. 

6.  Of  .29  to  six  places  of  decimals Ans.  .538516. 

The  square  root  of  a  whole  number  and  a  decimal,  may  be  found 

in  the  same  manner.  Thus,  the  square  root  of  2.5  is  the  same  as 
the  square  root  of  f§o',  which,  carried  out  to  6  places  of  decimals, 
is  1.581 138+. 

7.  Find  the  square  root  of  10.76  to  six  places  of  decimals. 

Ans.  3.260243. 

8.  Find  the  square  root  of  1.1025 Ans.  1.05. 

When  the  denominator  of   a  fraction  is  a  perfect  square,  its 

square  root  may  be  found  by  extracting  the  square  root  of  the 
numerator  to  as  many  places  of  decimals  as  are  required,  and  di- 
viding the  result  by  the  square  root  of  the  denominator.  Or,  by 
reducing  the  fraction  to  a  decimal,  and  then  extracting  its  square 

R  E  V I E  w. — 194.  How  do  you  find  the  approximate  square  root  of  a  frac- 
tion to  within  one  of  the  equal  parts  of  the  denominator?  How  do  you 
extract  the  square  root  of  a  decimal  ?  How  do  you  extract  the  square  root 
of  a  fraction,  when  both  terms  aro  not  perfect  squares  ? 


EXTRACTION  OF  THE  SQUARE  ROOT.  183 

root.     AVhen  the  denominator  of  the  fraction  is    not  a  perfect 
square,  the  latter  method  should  be  used. 

9.  Find  the  square  root  of  |  to  five  places  of  decimals. 
|/I=1.73205+,  1/4=2,  /|=ii7_3?-0_5+=.86602+. 

Or,  -|=.75,  and  i/775=.86602+. 

10.  Find  the  square  root  of  3§       Ans.  1.795054+. 

11.  Find  the  square  root  of  j\.      .....     Ans.  .661437+. 

12.  Find  the  square  root  of  3 { Ans.  1.802775+. 

13.  Find  the  square  root  of  5f Ans.  2.426703-f . 

14.  Find  the  square  root  of  ^ Ans.  .377964+. 

15.  Find  the  square  root  of  | Ans.  .935414+. 

16.  Find  the  square  root  of  2| Ans.  1. 527525+. 

EXTRACTION   OF  THE   SaUARE  ROOT  OF  MONOMIALS. 

Art  195. — From  the  principles  in  Art.  179,  it  is  evident,  that 
in  order  to  square  a  monomial,  we  must  square  its  coefficient,  and 
multiply  the  exponent  of  each  letter  by  2.     Thus, 

Therefore,  y9d'b*=Sab\     Hence,  the 

RULE, 

FOR    EXTRACTING    THE    SQUARE    ROOT    OF   A    MONOMIAL. 

Extract  the  square  root  of  the  coefficient,  and  divide  the  exponent 
of  each  letter  by  2. 

Since  +aX+«=+«^  and  — aX — a— -+a^ 

Therefore  v^a'^=+<x,  or  — a. 

Hence,  the  square  root  of  any  positive  quantity  is  either  plus, 
or  minus.  This  is  generally  expressed,  by  writing  the  double  sign 
before  the  square  root.  Thus,  ■j/4a'^=±2a,  which  is  read,  plus  or 
minus  2a. 

If  a  monomial  is  negative,  the  extraction  of  the  square  root  is 
impossible,  since  the  square  of  any  quantit}^  either  positive  or 
negative,  is  necessarily  positive.  Thus,  i/— 9,  \/ — 4a^  |/ — b,  are 
algebraic  symbols,  which  indicate  impossible  operations.  Such 
expressions  are  termed  imaginary  quantities.  They  occur,  in  at- 
tempting to  find  the  value  of  the  unknown  quantity  in  an  equation 
of  the  second  degree,  where  some  absurdity  or  impossibility  exists 
in  the  equation,  or  in  the  problem  from  which  it  was  derived. 
See  Art.  218. 

Review. — 195.  How  do  we  find  the  square  of  a  monomial?  How, 
then,  do  we  find  the  square  root  of  a  monomial  ?  What  is  the  sign  of  the 
square  root  of  any  positive  quantity  ?  Why  is  the  extraction  of  the  square 
root  of  a  negative  monomial  impossible  ?  Give  examples  of  algebraic  sym- 
bols that  indicate  impossible  operations.  What  are  they  termed?  Under 
■what  circumstances  do  they  occur? 


184  RAY'S   ALGEBRA,    PART    FIRST. 


Find  the  square  root  of  each  of  the  following  monomials. 

1.  4aV'.    .    .    .   Ans.  ±2ax, 

2.  9xy.    .    .      Ans.  ±3x/. 

3.  2f)aWc\  .    Ans.  ±5a6cl 

4.  ^Qa'Wx-.  .  Ans.  dzGa^^-'^x. 


if.   .   Ans.  dz4:mn'^f. 
6.  49a26*c8.     .     Ans.  ±7a6V. 


7.  625x22^     .    .  Ans.  d=.25xz\ 

8.  1156aV26.  Ans.  ±34axV. 


Since   ijj  =7XT=y2'  therefore,  W—-2=zfcr  ;   that  is,  /o  ^';t(i 

^Ae  square  root  of  a  monomial  fraction,  extract  the  square  root  of 
both  terms. 

9.  Find  the  square  root  of  -qt^ Ans.  ±oT- 

10.  Find  the  square  root  of -^(T-f^ Ans.  ±-f- — . 

EXTRACTIOIV    OF    THE    SftUARE    ROOT    OF   POLYAOMIALS. 

Art,  196. — In  order  to  deduce  a  rule,  for  extracting  the  square 
root  of  a  polynomial,  let  us  first  examine  the  relation  that  exists 
between  the  several  terms  of  any  quantity  and  its  square. 

{a-\-hY=d'+2ah+h''=d'+[2a^b)h. 
(a+6+c)'-'=a2+2a6+62+2a6'+26c+c2=a2+(2a+6)6+(2a+26 

(a+6+c+<^)'=a''+2a6+&'+2ac+2&c+c2+2acZ-l-2&(f+2cc?+^' 
=a'-^+(2a+&)6+(2a+26+c)c+(2a+26+2c+^)(i 

Or,  by  calling  the  successive  terms  of  a  polynomial  r,  i\  i^',  r'", 
we  shall  have  (r+r'+^-"+/")'=^'+ (2r+/)^''+ (2r+2r'+J-'0''" 
+  (2r+2/+2/'+/")^-"'- 

In  this  formula,  r,  /,  /',  ?•'",  may  represent  any  algebraic  quan- 
tities whatever,  either  whole  or  fractional,  positive  or  negative. 

Hence,  we  see,  that  the  square  of  any  polynomial  is  formed 
according  to  the  following  law: 

The  square  of  any  polynomial  is  equal  to  the  square  of  the  first 
term — plus  twice  the  first  term,  plus  the  second,  multiplied  by  the  sec- 
ond— plus  twice  the  first  and  second  terms,  plus  the  third,  multiplied 
by  the  third— plus  iioice  the  first,  second,  and  third  terms,  plus  the 
fourth,  midtiplied  by  the  fourth,  and  so  on.  Hence,  by  reversing 
the  operation,  we  have  the 

RULE, 

FOR   EXTRACTING    THE    SQUARE    ROOT    OF    A    POLYNOMIAL. 

1  st.  Arrange  the  polynomial  with  reference  to  a  certain  letter;  then 
find  the  first  term  of  the  root,  by  extracting  the  square  root  of  the 
first  term  of  the  polynomial ;  place  the  result  on  tlie  right,  and  sub- 
tract its  square  from  the  given  quantity. 


EXTRACTION"  OF  THE  SQUARE  ROOT.  185 


2d.  Divide  the  first  term  of  the  remainder,  by  double  the  part  of 
the  root  already  found,  and  annex  the  result  both  to  the  root  and  the 
divisor.  Multiply  the  divisor  thus  increased,  by  the  second  term  of 
the  root,  and  subtract  the  product  from  the  remainder. 

3d.  Double  the  terms  of  the  root  already  found,  for  a  partial  divi- 
sor, and  divide  the  first  term  of  the  remainder,  by  the  first  term  of 
the  divisor,  and  annex  the  residt  both  to  the  root  and  the  partial  divi- 
sor. Multiply  the  divisor  thus  increased,  by  the  third  term  of  the 
root,  and  subtract  the  product  from  the  last  remainder.  Then  pro- 
ceed in  a  similar  manner,  to  find  the  other  terms. 

Re  SI  A  UK. — In  the  course  of  the  operations  on  any  example,  when  we 
find  a  remainder,  of  which  the  first  term  is  not  exactly  divisible  by  double 
the  first  term  of  the  root,  we  may  conclude  that  the  polynomial  is  not  a  per- 
fect square. 

EXAMPLES. 

1.  Find  the  square  root  of  rM-'^r/+r'2+2r/'+2>V'+r'^ 
^.2_^2r/+/^-f2r/^H-2//^+/^^-1r+/+^-'',  root. 

T^ 

2r+/  j2r/+^'" 

|2;V+r^^ 

2,.+2/+r''  ]2r/'-f-2/V'+/'2 

2r/'-|-2//^'+7-"^ 


The  square  root  of  the  first  term  is  r,  which  we  write  as  the  first 
term  of  the  root.  We  next  subtract  the  square  of  r  from  the 
given  polynomial,  and  dividing  the  first  term  of  the  remainder 
2ry,  by  2r,  the  double  of  the  first  term  of  the  root,  the  quotient  is 
r,  the  second  term  of  the  root.  We  next  place  /  in  the  root,  and 
also  in  the  divisor,  and  multiply  the  divisor  thus  increased,  by  /, 
and  subtract  the  product  from  the  first  remainder.  We  then 
double  the  terms  r+/,  of  the  root  already  found,  for  a  partial  divi- 
sor, and  find  that  the  quotient  of  2rr",  the  first  term  of  the  remain- 
der, divided  by  2r,  the  first  term  of  the  divisor,  is  r",  the  third 
term  of  the  root.  Completing  the  divisor,  multiplying  by  /',  and 
subtracting,  we  find  there  is  nothing  left. 

Note.— The  first  remainder  consists  of  all  the  terms  after  r^,  and  the 
second,  of  all  after  r"^.  It  is  useless  to  bring  down  more  terms  than  have 
corresponding  terms  in  the  quantity  to  be  subtracted. 

Review.— 196.  What  is  the  square  of  a-\-hl  Of  «-f-i-fc?  Of  <t-\  b 
_j_g_|_(^?  Q^  r^r''\-r" -{-)•" "i  What  may  r,  r',  <Sbc.,  represent  ?  Accord- 
ing to  what  law  is  the  square  of  any  polynomial  formed?  By  reversing 
this  law,  what  rule  do  we  obtain,  for  extracting  the  square  root  of  a  poly- 
nomial ?     When  may  we  conclude  that  a  polynomial  is  not  a  perfect  square  ? 

16 


186  RAY'S   ALGEBRA,    PART   FIRST. 

2.  Find  the  square  root  of  the  polynomial  25xy — 24xy^ — 12x^y 
+4x*+l6>/\ 

Arranging  the  polynomial  with  reference  to  x,  we  have 
4x*—  1 2x3?/+25xy — 24x/+ 1 67/\2x'-Sx7/+4if,  root. 

4x* 

'ix'^—Sxi/ — 1 2x'i/-h2DxY 

|— 12x^//+  9xy 

4x'~Qxt/+4if\  1 6x2?/2-24x?/3+ 1 6/ 
|lGxy-24x?/+16// 

It  is  easily  seen,  that  the  operation  is  analogous  to  that  of  ex- 
tracting the  square  root  of  whole  numbers. 

Find  the  square  root  of  the  following  polynomials. 

3.  x2H-4x+4 Ans.  x+2. 

4.  4x2-12x+9. Ans.  2x-3. 

5.  xy — 8x//+16 Ans.  xy — 4. 

6.  4a'^x-+25?/V — 20axyz Ans.  2ax — 5?/2. 

7.  x*+4x3+6x2+4x+l Ans.  x2+2x+l. 

8.  4x*— 4x-"'+13x2-Gx+9 Ans.  2x2— x+3. 

9.  9^— r2?/='+34//-20y+25 Ans.  3y-2y+5. 

10.  aV+6a''6V— 4a''6x3— 4«Z>^x+^*.     .    .  Ans.  a?x''—2ahx-^h\ 

1 1 .  1— 4x+ 1  Ox^— 20x'+25x*-24x»+ 1 6x«. 

Ans.  1— 2x+3x"^— 4x1 

12.  a^—Qw>x+ 1 5aV— 20aV+ 1 5aV -6ax5+x«. 

Ans.  a"* — 3a'^x-L3ax^— x^. 

13.  x^+ax+Jtf^ Ans.  x-\-\a. 

14.  x'—2x+\+2xy—2y-\-y' Ans.  x+y— 1. 

15.  x(x+l)(x+2)(x+3)+"l Ans.  x2+3x+l. 

Art.  197. — The  following  remarks  will  be  found  useful. 

1st.  No  binomial  can  he  a  perfect  square;  for,  the  square  of  a 
monomial  is  a  monomial,  and  the  square  of  a  binomial  is  a  trino- 
mial. Thus,  d^-\-h'^  is  not  a  perfect  square;  but  if  we  add  to  it 
2ab,  it  becomes  the  square  of  a-\-h ;  and  subtracting  from  it  2ab, 
it  becomes  the  square  of  a — h. 

2d.  In  order  that  a  trinomial  may  be  a  perfect  square,  the  two 
extreme  terms  must  be  perfect  squares,  and  the  middle  term  the 
double  product  of  the  square  roots  of  the  extreme  terms.  Hence, 
to  obtain  the  square  root  of  a  trinomial  when  it  is  a  perfect  square, 
extract  the  square  roots  of  the  two  extreme  terms,  and  unite  them  by 
the  sign  plus  or  minus,  according  as  the  second  term  is  plus  or  minus. 

Review. — 197.  Why  can  no  binomial  be  a  perfect  square  ?  Give  an 
example.  What  is  necessary,  in  order  that  a  trinomial  may  be  a  perfect 
(square  ?  When  a  trinomial  is  a  perfect  square,  how  may  its  square  root  be 
found?     Give  an  examnk>. 


RADICALS  OF  THE  SECOND  DEGREE.  187 

Thus,  4a^ — 12ac-f9c^  is  a  perfect  square,  since  |/4a'^=2a, 
y'9?=Sc,  and  +2aX-3cX2=-12ac.  But  9x-'+l2xi/+l67f,  is 
not  a  perfect  square  ;  since  y  dx-^^Sx,  y\Qi/z::^Ay,  and  3xX4?/X2 
=24xy,  which  is  not  equal  to  the  middle  term  \2xy. 


RADICALS    OF    THE    SECOND    DEGREE, 

Art.  198. — From  the  rule  Art.  195,  it  is  evident,  that  ivlien  a 
monomial  is  a  •perfect  square,  its  numeral  coefficient  is  a  perfect 
square,  and  the  exponent  of  each  letter  is  exactly  divisible  by  2. 
Thus,  4a^  is  a  perfect  square,  while  5a^  is  not  a  perfect  square, 
because  the  coefficient,  5,  is  not  a  perfect  square,  and  the  expo- 
nent, 3,  is  not  exactly  divisible  by  2. 

When  the  exact  division  of  the  exponent  can  not  be  performed, 
it  may  be  indicated,  by  writing  the  divisor  under  it,  in  the  form 

of  a  fraction.     Thus,  \/a^  may  be  written  a^. 

Since  a  is  the  same  as  a^  the  square  root  of  a  may  be  expressed 

thus,  a^.  For  this  reason,  the  fractional  exponent,  I,  is  used  to 
indicate  the   extraction  of  the  square  root.     Thus,  ^ d^-\-2ax-\- x^ 

L  —  ?    .      . 

and  (a^H-2ax+,'C^)-,  also  1/4  and  4~,  indicate  the  same  operation  ; 

the  radical  sign,  |/,  and  the  fractional  exponent,  i,  being  regarded 
as  equivalent. 

Quantities  of  which  the  square  root  can  not  be  exactly  ascer- 
tained, are  termed  radicals  of  the  second  degree.  They  are  also 
called,  irrational  quantities,  or  surds.     Such  are  the  quantities  ya, 

|/2,  ay^b,  and  5>/3.     Or,  as  they  may  be  otherwise  written,  a^, 

1       J  J 

'2'^,  ab'^,  and  5(3)^.  The  quantity  which  stands  before  the  radi- 
cal sign,  is  called  the  coefficient  of  the  radical.  Thus,  in  the 
expressions  a\/b,  and  3|/5,  the  quantities  a  and  3  are  called 
coefficients. 

Two  radicals  are  said  to  be  similar,  when  the  quantities  under 
the  radical  sign  are  the  same  in  both.  Thus,  3i/2  and  l\/2  are 
similar  radicals;  so,  also,  are  b\/a  and  '2c\/a. 

Two  radicals  that  are  not  similar,  may  frequently  become  so, 
by  simplification.     This  gives  rise  to 

Re  VIE  w. — 198.  When  is  a  monomial  a  perfect  square?  Give  an  ex- 
nraple.  How  may  the  square  root  of  a  quantity  be  expressed,  without  the 
radical  sign  ?  What  are  radicals  of  the  second  degree  ?  What  are  radicals 
otherwise  called?  What  is  the  coefficient  of  a  radical?  When  are  two 
radicals  similar  ? 


188  RAY'S   ALGEBRA,    PART    FIRST. 

REDUCTION   OF    RADICALS   OF    THE   SECOXD   DEGREE. 

Art.  199. — Reduction  of  radicals  of  the  second  degree,  con« 
sists  in  changing  the  form  of  the  quantities  without  altering  their 
value.     It  is  founded  on  the  following  principle. 

The  square  root  of  the  product  of  two  or  more  factors,  is  equal  to 
the  product  of  the  square  roots  of  those  factors. 

That  is,  ■/ ab=y^ ay(_i/ b ;  which  is  thus  proved: 

(/^^a6_       __         _         ____       

and  (i/aX;/^)— i/«Xi/^Xi/«Xi/6=/aX>/«X/6X|/6=:a6. 

Hence,  \/ab  and  |/aX|/6  are  equal  to  each  other,  since  the 
square  of  each  is  equal  to  ab. 

From  this  principle,  we  have  /36=/4X9=2X3,  |/144 
-1/9X16-3X4. 

Any  radical  of  the  second  degree,  can  be  reduced  to  a  simpler 
form  when  it  can  be  separated  into  factors,  one  of  which  is  a  per- 
fect square. 

Thus,  y'T2=|/4X3-v/4Xv/3-2/3  __ 
y'  a^b=i/  d^X.ab=:\/  a^X,\/  ab^=a\/  ab 
V  27^V=T/yaVX37r— /  y^^X  V'Sa=Sacy'3a. 

From  the  preceding  illustrations,  we  derive  the 

RULE, 

FOR  REDUCING  A  RADICAL  OF  THE  SECOND  DEGREE  TO  ITS  SIMPLEST  FORM. 

1st.  Separate  the  quantity  to  be  reduced,  into  tivo  ^mrts,  one  of 
lohich  shall  contain  all  the  factors  that  are  perfect  squares,  and  the 
other  the  remaining  factors. 

2d.  Extract  the  square  root  of  the  part  that  is  a  perfect  square,  and 
prefx  it  as  a  coifficient,  to  the  other  part  placed  under  the  radical  sign. 

To  determine  if  any  quantity  contains  a  numeral  factor  that  is 
a  perfect  square,  ascertain  if  it  is  exactly  divisi])le  by  either  of  the 
perfect  squares,  4,  9,  16,  25,  86,  49,  64,  81,  100,  121,  144,  &c. 
If  not  thus  divisible,  it  contains  no  factor  that  is  a  perfect  square, 
and  the  numerical  factor  can  not  be  reduced. 

Reduce  each  of  the  following  radicals  to  its  simplest  form. 

1.  yW\  Ans.  2ai/27 


2.  |/12a-l  Ans.  2aj/3a. 

3.  yiQd'b.  Ans.  4a/a6. 

4.  V\Sa*h'c\     A.  3a2k'|,/26c. 


5.  / 20(2^6 V.  A.  2abc\/  babe. 

6.  3/24^^.  Ans.  Qa'cy^ 

7.  4v/27aV.  A.  Uacy/Sac. 

8.  7T/28aV^  A.  Ma'c^/la. 


9.  /32aW.     Ans.  4a^bcy2. 

10.  i/40aW.    A.  2abcyi0bc. 

1 1 .  i/44a^b'c.    A.  2a''bi/ 1  l«6c. 

12.  ■/45aW.    Ans.  3a'6VV5. 

13.  y48aH^.       A.  4a*b'cyS. 

14.  i/75aW.__  A.  5a6c|/3a6c. 

15.  |/128a'^6V.     A.  Sa^b^ci/2^ 

16.  |/243a=^6^c.       A.  9aby/^ac. 


RADICALS  OF  THE  SECOND   DEGREE.  189 

In  a  similar  manner,  polynomials  may  sometimes  be  simplified. 

Thus,  i/[2a:'—4^'b+2ab'')=^/2a[d'—2ab-j-b-')={a—b)  ^2a. 

A  fractional  radical  of  the  second  degree  may  be  reduced  to  its 
simplest  form,  by  the  same  rule,  by  first  multiplying  both  terms 
by  any  quantity  that  will  render  the  denominator  a  perfect  square; 
t^eparating  the  fraction  into  two  factors,  one  of  which  is  a  perfect 
f  quare,  then  extracting  the  square  root  of  the  square  factor,  and 
] 'lacing  it  before  the  other  factor  placed  under  the  radical  sign. 

17.  Reduce  -^Z;^  to  its  simplest  form, 

vJ=vixi=v'i=v'^Q=v%y<v'^^lv'^'  Ans. 

Reduce  the  following  fractional  radicals  to  their  simplest  forms. 


18.  ]/i.  Ans.  1/15T 

19.  v'i-  Ans.  jyTi: 

20.  i/]f  Ans.  f/S: 

21.  vtl-  ^"s-  IV^' 


22.  91/^"^.  Ans.  4/3. 

23.  W'J^-  Ans.  3/TO. 

24.  lOi//"  Ans.  VQ. 

25.  IVlh-  Ans.  J,i/2r 


•Z6 


Since  a^y  a\  and  2i/3=]/4X/3=]/4X3=]/12,  it  is  obvi- 
ous, that  any  quantity  may  be  reduced  to  the  form  of  a  radical 
<.f  the  second  degree,  by  squaring  it,  and  placing  it  under  the 
radical  sign.  By  the  same  principle,  the  coefficient  of  a  radical 
may  be  passed  under  the  radical  sign. 

26.  Reduce  5  to  the  form  of  a  radical  of  the  second  degree. 

Ans.  1/25: 

27.  Reduce  2a  to  the  form  of  a  radical  of  the  second  degree. 

Ans.  \/4.d\ 

28.  Express  the  quantity  3]/ 5,  entirely  under  the  radical. 

Ans.  |/45. 

29.  Pass  the  coefficient  of  the  quantity  3ci/2c,  under  the  radical. 

Ans.  |/T8?^ 

30.  Pass  the  coefficient  of  the  quantity  byS,  under  the  radical. 

Ans.  |/75. 

ADDITIOIV    OF    RADICALS    OF    THE    SECOND    DEGREE. 

Art.  200.— 1.  What  is  the  sum  of  3^/2  and  5/2? 
It  is  evident,  that  3  times  and   5  times  any  certain  quantity 
must  make  8  times  that  quantity,  therefore 

3/2+5i/2-=8i/27 
In  the  same  manner,  i/'2H-|/8=i/2+2]/2— 3/2. 
2.  What  is  the  sum  of  2/3  and  5/7  ? 

Since  dissimilar  quantities  can  not  be  collected  into  one  sum,  we 
can  only  add  these  expressions  by  placing  the  sjgn  of  ^addition 
between  them;  that  is,  the  sum  of  2/3  and  5/7=2;/ 3-f  5/7. 


190  RAY'S    ALGEBRA,    PART  FIRST. 

Hence,  the 

RULE, 

FOR    THE    ADDITION    OF    RADICALS    OF    THE    SECOND    DEGREE. 

1st.  Reduce  the  radicals  io  their  simj^lest  Jorm. 

2d.  Then,  if  the  radicals  are  similar,  prejix  the  sum  of  their  coef- 
ficients to  the  common  radical;  but,  if  the?/  are  not  similar,  con- 
nect them  by  their  proper  signs. 

Find  the  sum  of  the  radicals  in  each  of  the  following  examples. 

3.  v^S^nd  ]/Ta_ Ans.  5v  2. 

4.  i/~12  and  /27. Ans.  5i/3. 

5.  /20  and  /80. Ans.  G/S^ 

6.  i/24and_v/150.  ^ Ans.  Ti/G. 

7.  >/8,  ;/32,  and  /50. .     Ans.  lli/2. 

8.  >/40,  t790,  and  /250. Ans.  lOi/lO. 

9.  V'ZSd'b'  and  /H2a'-'6' Ans.  (Sab^Y. 

10.  >/75a^c,  andi/MTa-'c Ans.  12ai/3c. 

11.  |/J  and/~3^ • Ans.  /^v/3. 

12.  /iandv/^ Ans.  ^§1/5. 

13.  V.V  and  >/8 Ans.  5|/^2. 

14.  2/1  and  3v/T2 Ans.  ly% 

15.  jy^and  |/2. Ans.  v/2. 

16.  3v/|  and  7i/|| Ans.  ^J/G. 

17.  -/iSo^^x-  and  |/126'^a; Ans.  (4«c+25)i/3^'. 

18.  Find  the  sum  of  \/{2d^ — ^d^c-\-)Zac')  and  

l/(2a='+4V6'+2a6^).  Ans.  2a>/2a.' 

19.  Find  the  sum  of  ^ a-^x-^-^ ax'^3?-^y\a-\-xf,  

Ans.  (l+a+2a:)/a+x. 

SUBTRACTIOiX    OF  RADICALS   OF   THE  SECOi\D  DEGREE. 

Art.  '201.-1.  Take  3/2  from  5/2. 

It  is  evident  that  5  times  any  quantity  minus  3  times  the  quan- 
tity, will  be  equal  to  2  times  the  quantity,  therefore 
5/2-3/2=2/2.  _ 

In  the  same  manner,  /8 — /2=^2/2 — /2=^/2. 

Review.  —  199.  In  what  does  reduction  of  radicals  of  the  second 
degree  consist?  On  what  principle  is  it  founded?  Prove  this  principle. 
What  is  the  rule  for  the  reduction  of  a  radical  of  the  second  degree  to  its 
simplest  form?  How  do  you  determine  if  any  quantity  contains  a  numer- 
ical factor  that  is  a  perfect  square?  How  may  a  fractional  radical  of  the 
second  degree  be  reduced  to  its  simplest  form  ?  200.  What  is  the  rule  for 
the  addition  of  radicals  of  the  second  degree? 


RADICALS  OF  THE  SECOND  DEGREE.  101 

If  the  radicals  are  dissimilar,  it  is  obvious  that  their  difference 
can  only  be  indicated.  Thus,  if  it  be  required  to  take  3]/a  from 
5|/6,  the  difference  would  be  expressed  by  5\/l)—S\/a. 

From  these  illustrations,  we  derive  the 

RULE, 

FOR    THE    SUBTRACTION    OF    RADICALS    OF    THE    SECOND    DEGREE. 

1  St.  Reduce  the  radicals  to  their  simplest  form ;  then  subtract  their 
coefficients,  and  prejix  the  difference  to  the  common  radical. 

2d.  If  the  radicals  are  not  similar,  indicate  their  difference  by  the 
proper  sign. 

EXAMPLES. 

2.  y/lS— v/2. Ans.  V2. 

3.  >/45a^— |/5a^ Ans.  2av/5. 

4.  v/546— v/66.  .    . Ans.  2/66. 

5.  v/Tr2aV— i/28a_V Ans.  2acj/7. 

6.  v/276^3_^26V Ans.  bc^Uc^ 

7.  ■\/S6aF—\/4a^.    . Ans.  4aV^ 

8.  1/49^6^^—- i/25a6V^. Ans.  2bcy^ab. 

9.  /TOOo^c— i/10a'^6=^c Ans.  3a6|/10a6c. 

10.  5a/27-3aT/4a Ans.  3av/3. 

11.  2v/|-3]/i Ans^. 

12.  vtrv^j:    ^°«-  iW^^ 

13.  i/12-|/| Ans.  -3y3. 

14.  3;/^— /2 Ans.  1/2; 

15.  -,/|l/g^.    .    .    .    ._ Ans.  fi/O; 

16.  From  v/4a"^x  take  a/x-^     ....    .    .    .  Ans.  {2a—ax)yx. 

17.  From  \/3m'^x-\-Qmnx-\-Sn^x  take  ^/Sm^x — {jmnx-JrSn^x.  

Ans.  2ni/3a.-. 

MULTIPLICATION    OF   RADICALS  OF  THE  SECOND  DEGREE. 

Art.  202.— Since   i/ab=\/aXV'^y   therefore   |/aX-/6=]/a6. 
See  Art.  199. 
Also,  ai/bXc\/d=-(^XcXVbXVd=ac^bd. 
From  which  we  have  the 

RULE, 
FOR   THE    MULTIPLICATION    OF    RADICALS    OF    THE    SECOND    DEGREE. 

1st.  Multipli/  the  quantities  under  the  radical  sign  together,  and 
place  the  result  under  the  radical. 

2d.  If  the  radicals  have  coefficients,  place  their  product  as  a  coef- 
ficient before  the  radical  .tign. 


:92  RAY'S   ALGEBRA,    PART   FIRST. 


EXAMPLES. 

1.  Find  the  product  of  -p/O  and  |/8. 

/6Xi/'8=/48=/r6X3=:4>/3.    Ans. 

2.  Find  the  product  of_2v/14  and  3|/2.        _  _ 
2i/T4X3i/2=6/28=6v/4X7-=6X2i/7=12i/7.    Ans. 

3.  Find  the  product  of  /Sjind  1/2.  _. Ans.  4. 

4.  Find  the  product  of  2\/a  and  3i/a Ans.  6a. 

5.  Find  the  product  of  y'27  and  |/3 Ans.  9. 

6.  Find  the  product  of  3/2  and  2]/3 Ans.  6/6^ 

7.  Find  the  product  of  3/3  and  2/3 Ans.  18. 

8.  Find  the  product  of  /6  and  |/15.     ....     Ans.  3/10". 

9.  Find  the  product  of  2/l5  and  3/3^5.      .    .  Ans.  30/21. 

10.  Find  the  product  of  \/ ci/'O^'c  and  y'aOc Ans.  d^b^c. 

11.  Find  the  product  of  /-I  and  /  3 Ans.  1. 

12.  Find  the  product  of  /I  and  i/l". Ans.  j\V^- 

13.  Find  the  product  of  2w^  and  3 W,-T.    .    .    .     Ans.  -^/2. 

When  two  polynomials  contain  radicals  of  the  second  degree, 
they  may  be  multiplied  together,  in  the  same  manner  as  in  multi- 
plication of  polynomials,  Art.  72,  attending,  at  the  same  time,  to 
the  directions  contained  in  the  preceding  rule. 

14.  Find  the  product  of  2+/2  and  2-/2 Ans.  2. 

15.  Find  the  product  of  lH-y''2  and  1 — /2.  .    .    .       Ans. — 1. 

16.  Find  the  product  of  -/x+l^  by  /x— 2.      .    .  Ans.  \/x^ — 4. 

17.  Find  the  product  of  \/ a-\-x  by  y  a-\-x Ans.  a-\-x. 

18.  Find  the  product  of  y^ ab-\-hx  by  \/ab-~bx.  A.  \/ d^b'^—b'^xK 

19.  Find  the  product  of  /x+2~by  /x+3T    Ans.  /xM^5^6.' 
Perform  the  operations  indicated  in  the  following  examples. 

20.  (c/^+(V^)X(c/a— r?/6j Ans.  c'^a—d'b. 

21.  (7+2/6)X(9— 5/6).    .    .    .    .    .    .    .    .  Ans.  3— 17/6. 

22.  (/a+x+/a — x)(/a+x — /« — x) Ans.  2x. 

23.  {x-{-2-i/ ax-{-a){x — 2-/ax+a) Ans.  x^—2ax-\-dK 

24.  (a;2— x/2+l)(a;'^+a:/^+l) Ans.  x*+l. 

DIVISIO!V  OF   RADICALS  OF  THE  SECOND  DEGREE. 

Art.  203. — Since  Division  is  the  reverse  of  Multiplication,  and 

since  \/ay(i/b=^i/ab,  therefore  /a6-i-/a=^/^=/6. 

Ke  VIE  w. — 201.  What  is  tho  rule  for  the  subtraction  of  radicals  of  the 
eecond  degree  ?  202.  What  is  tho  rule  for  the  multiijlication  of  radicals  of 
the  second  degree  ?     On  what  principle  does  it  depend  ? 


RADICALS  OF  THE  SECOND   DEGREE.  193 

Also,  since  a\/ b'Xci/ d=aci/ bd,  therefore  acy^6d-T-a]/6= — — :=. 

^V  b 

ac 


— ^/^-=ci/<i.     Hence,  the 


RULE, 

FOR   THE   DIVISION   OF   RADICALS   OF   THE   SECOND   DEGREE. 

1st.  Find  the  quotient  of  the  parts  under  the  radical,  and  place  it 
under  the  common  radical. 

2d.  If  the  radicals  have  coefficients,  divide  the  coefficient  of  the 
dividend  by  that  of  the  divisor,  and  prefix  the  result  to  the  common 
radical. 

Note. — When  a  radical  quantity  has  no  coefficient  prefixed,  its  cogffi- 
ciont  is  understood  to  be  1.     Thus,  ^2  is  the  same  as  l|/2.     See  Art.  32. 


EXAMPLES. 

1.  Divide  8^/72  by  2/6. 

:|/y  =41/12=41/4X3^81/3:    Ans. 


8i/72_8  /,,_ 


2/6 

2.  Divide  /54by  /  6.  ^_. Ans.  3. 

3.  Divide  6/54  by  3/27 Ans.  2/2. 

4.  Divide  6/28  by  2/7 Ans.  6. 

5.  Divide /160_by /Si Ans.  2/5. 

6.  Divide  15/378  by  5/6.     . Ans.  9/7. 

7.  Divide  /a^  by  /a.    . Ans.  a. 

8.  Divide  ab\/aV)^  by  6/«6. Ans.  a^b. 

9.  Divide  a  hx  /« Ans.  /a. 

10.  Divide  a/6  by  t*/^ Ans. -^/-^,  or— /6c/. 

11.  Divide  ^1  by  ^1 Ans.^^,orl/^. 

12.  Divide  /i  by  /J Ans.  ^/6. 

13.  Divide /|  by /^l Ans.  1^. 

14.  Divide  I /T8  by  1-/2 Ans.  4. 

15.  Divide  ly'l  by  ^v  J- Ans.  f/S". 

16.  Divide  ^/|  by /2+3/'^ Ans.  J^. 

Art.  204.— To  reduce  a  fraction  whose  denominator  is  either  a 
monomial  or  a  binomial  containing  radicals  of  the  second  degree, 
to  an  equivalent  fraction  having  a  rational  denominator. 

Review. — 203.  "What  is  the  rule  for  the  division  of  radicals  of  the 
second  degree  ?     On  what  principle  does  it  depend  ? 

17 


194  RAY'S   ALGEBRA,    PART   FIRST. 


When  the  fraction  is  of  the  form  -^,  if  we  multiply  both  terms 

hy  \/b,  the  denominator  will  become  rational.     Thus, 
«__  aXi/ft  _ai/b 

Since  the  sum  of  two  quantities,  multiplied  by  their  difference, 
is  equal  to  the  difference  of  their  squares;  if  the  fraction  is  of  the 

form  =L,  and  we  multiply  both  terms  by  b — ]/c,  the  denomin- 

b+i/c 
ator  will  be  made  rational,  since  it  will  be  6^ — c.     Thus, 

a         b — |/c ab — a-^/c 

■  X' 


h-{-yc     b—i/c  ^'—^ 

For  the  same  reason,  if  the  denominator  is  b — |/c,  the  multi- 
plier will  be  b-\-\/ c.  If  the  denominator  is  \^b+\/c,  the  multi- 
plier will  be  \/b — y'c,  and,  if  the  denominator  is  |/6 — \/c,  the 
multiplier  will  be  y'6+|/c. 

These  different  forms  may  be  embraced  in  the  following 

GENERAL   RULE. 

If  the  denominator  is  a  monomial,  multiply  both  terms  by  the  rad- 
ical quantity;  but,  if  it  is  a  binomial,  multiply  both  terms  by  the 
given  binomial  with  the  sign  of  one  of  its  terms  changed,  and  the 
denominator  will  be  rational. 

Reduce  the  following  fractions  to  equivalent  fractions,  having 
rational  denominators. 


4-  ^^^     An..  ,'T(6+y  3). 


1.  -4.  Ans.  l4?=4i/2. 
v^2  ^_ 

2.  ^.  Ans.  ^^\VQ. 
>/3  ^ 

3.  — U=.  Ans.  2-1/3. 

2+V/3 

R  E  M  A  R  K. — The  utility  of  these  transformations,  consists  in  diminishing 
the  amount  of  calculation,  necessary  to  obtain  the  numerical  value  of  a 
fractional  radical  to  any  required  degree  of  accuracy. 

Thus,  suppose  it  is  required  to  obtain  the  numerical  value  of  the  fraction 

,  true  to  six  places  of  decimals. 

|/2 

If  we  make  the  calculation  without  rendering  the  denominator  rational, 
it  will  be  found,  that  we  must  first  extract  the  square  root  of  2,  to  seven 

R  E  V  I  E  \v. — 204.  When  the  denominator  of  a  fraction  is  either  a  mono- 
laial  or  a  binomial,  containing  radicals  of  the  second  degree,  how  may  it  be 
reduced  to  a  fraction  having  a  rational  denominator  ? 


SIMPLE  EQUATIONS  CONTAINING  RADICALS.      195 

places  of  decimals,  and  then  divide  1  by  this  result.  But  if  we  render  the 
denominator  rational,  the  calculation  merely  consists  in  finding  the  square 
root  of  2,  and  then  dividing  by  2.  The  work  by  the  latter  method,  requires 
only  about  half  the  labor  of  that  by  the  former.  Besides,  the  operator  feels 
certain,  if  he  has  made  no  mistake,  that  the  last  figure  of  his  result  is  cor- 
rect. "Whereas,  by  the  other  mode,  as  the  divisor  is  too  small,  the  quotient 
figures  soon  become  too  large.  Thus  in  this  example,  if  we  use  seven  deci- 
mals for  a  divisor,  the  seventh  figure  of  the  quotient  is  too  large ;  if  wo 
only  use  six  places  of  decimals,  the  sixth  figure  will  be  erroneous. 

7.  Find  the  numerical  value  of  the  fraction  — ^. 

Ans.  1.3416407+. 

Q 

8.  Find  the  numerical  value  of  the  fraction  -=z z=^. 

l/5— v2 

Ans.  3.650281 +. 

v/2 


9.  Find  the  numerical  value  of  the  fraction 


/5— v/3 

Ans.  2.805883+. 


R  E  M  A  R  K. — It  is  proper  to  notice,  that  the  signs  y  and  |/  ,  when 

applied  to  a  monomial,  both  have  the  same  meaning.  There  is  a  want  of 
uniformity  among  the  best  writers,  in  the  manner  of  making  the  radical 
sign  before  a  monomial. 

SIMPLE  EQUATIONS   COx\TAI\IIVG   RADICALS    OF    THE    SECOIVD 
DEGREE. 

Note  to  Teachers  . — This  part  of  the  subject  of  Equations  of  the 
Firct  Degree,  could  not  be  treated  till  after  Radicals.  It  may  be  omitted 
entirely  by  the  younger  class  of  pupils. 

Art,  205*  —  In  the  solution  of  questions  involving  radicals, 
much  will  depend  on  the  judgment  of  the  pupil;  but  the  easiest 
processes  can  only  be  learned  from  practice,  as  almost  every  ques- 
tion can  be  solved  in  several  ways. 

The  following  directions  will  be  frequently  found  useful. 

1st.  When  the  equation  contains  one  radical  expression,  trans- 
pose it  to  one  side  of  the  equation,  and  the  rational  terms  to  the 
other  side ;  then  involve  both  sides  to  a  power  corresponding  to 
the  radical  siscn. 


Thus,  if  we  have  the  equation  -[/{x — 1) — 1=2,  to  find  x. 
Transposing,  ],/(a: — 1)=3 
Squaring,  x — 1  =9,  from  which  a;=10. 

2d.  When  more  than  one  expression  is  under  the  radical  sign, 
the  operation  must  be  repeated. 


196  RAY'S   ALGEBRA,    PART   FIRST. 


Thus,  a-i-x=i/{d^-^xi/c'^-\'x'^),  to  find  x. 
Squaring,  a'^-}-2ax-\-x^=a^-\-xi/ c'^-jrx\ 


Reducing  and  dividing  by  x,  2a-\-x=-\/ c^-\-x^. 
Squaring,  4a'^-}-4ax-]-x^=c^-\'X^. 

Avhence  x= — . 

4a 

3d.  When  there  are  two  radical  expressions,  it  is  generally  bet- 
ter to  make  one  of  them  stand  alone  on  one  side,  before  squaring. 
Thus,  |/(a;— 5)— 3=4— /(x-— 12y,  to  find  x. 
Transposing,  |/(x— 5)=^7 — ^ [x — 12). 
Squaring,  a;— 5=49— 14>/(a;— r2)4-x— 12. 
Reducing  and  transposing,  14]/ (a: — 12)=42. 
Dividing,  /(x— 12)=3. 
Squaring,  x — 12=9,  from  which  ic=21. 

EXAMPLES    FOR    PRACTICE. 

1.  |/(^+3)+3=7 Ans.  a;=13. 

2.  x+T/(x-+ir)=ll Ans.  a:=5. 

3.  >/T^'+t/^— 1)=3 Ans.  x=10. 

4.  •/x(rt+x-)=a — X Ans.  x=^. 

5.  -Z"^— 2=i/'f^^)~. Ans.  x=9. 

6.  a:H-/x^^=7 Ans.  a:=4. 

7.  2+v/3x=/5^T4 Ans.  a:=12. 

8.  t/^7=6— r/x— 5 Ans.  x=9. 

_         _  25a 

9.  y'x— a=|/x— J^l/a Ans.  ic=-T7T  • 

10.  /x+225-|/x=424— 11=0 Ans.  x=1000. 

11.  x+T/2ax+x^=a Ans.  x=|a. 

12.  i/x+«— j/ic— a=|/a Ans.  ic=x* 


13.  i/x+12=24-i/a; Ans.  x=4. 

14.  /8+x=2v/T+x— |/^ Ans.  x=i. 

1 9  

15.  1/5x4- -^=r:z^=T/5x+6 Ans.  x=|. 

/5x+G 

16.  /^-aJ^^^~^1- Ans.  x=23. 

4+i/x 


17.  Va_ 

I.  \a-\-\/ax=\/a—\a — j/ax Ans.  x^^ti. 


=-l-l/4x^+a;+i/9x"^+12x=l+x Ans.  x- 

18. 


EQUATIONS  OF  THE  SECOND  DEGREE.  197 

19.  6(i/x+/6)=a(/^-/6) Ans.  x=-}^^l 

20.  y/x-^y^ax—a—l Ans.  x—{ya—\Y' 


CHAPTER  VII. 

EQUATIONS    OF    THE     SECOND    DEGREE. 

Art.  206. — An  Equation  of  the  Second  Degree  (See  Art.  148), 
is  one  in  which  the  greatest  exponent  of  the  unknovrn  quantity 
is  2.  Thus^  x'^-^Q,  and  5x^+3a;=26,  are  equations  of  the  second 
degree. 

An  equation  containing  two  or  more  unknown  quantities,  in 
which  the  greatest  exponent,  or  the  greatest  sura  of  the  exponents 
of  the  unknown  quantities,  is  2,  is  also  an  equation  of  the  second 
degree.  Thus,  xij=Q,  x^-{-xy=^S,  a:?/+a;+y=l  1,  are  equations  of 
the  second  degree. 

Equations  of  the  Second  Degree,  are  frequently  denominated 
Quadratic  Equations. 

Art.  soy. — Equations  of  the  second  degree  are  of  two  kinds — • 
incomplete  and  complete. 

An  incomplete  equation  of  the  second  degree,  is  of  the  form 
ax'^-—b,  and  contains  only  the  second  power  of  the  unknown  quan- 
tity, and  known  terms.  Thus,  x^=9,  and  Sx^ — 5x^=12,  are  in- 
complete equations  of  the  second  degree. 

An  incomplete  equation  of  the  second  degree,  is  frequently 
denominated  a  pinx  quadratic  equation. 

A  complete  equation  of  the  second  degree,  is  of  the  form 
ax^-{-bx=c,  and  contains  both  the  first  and  second  powers  of  the 
unknown  quantity,  and  known  terms.  Thus,  3x'^+4a;=20,  and 
ax^ — bx'^-'rdx — ex=f — g,  are  complete  equations  of  the  second 
degree. 

A  complete  equation  of  the  second  degree,  is  frequently  denom- 
inated an  affected  quadratic  equation. 

Revieav. — 206.  What  is  an  equation  of  the  second  degree?  Give  ex- 
amples. If  an  equation  contains  two  unknown  quantities,  when  is  it  of  the 
second  degree  ?  Give  examples.  207.  How  many  kinds  of  equations  of 
the  second  degree  are  there?  What  are  they?  What  is  the  form  of  an 
incomplete  equation  of  the  second  degree ?  What  does  it  contain?  Givo 
an  example.  What  is  the  form  of  a  complete  equation  of  the  second 
degree  ?  What  does  it  contain  ?  Givo  an  example.  What  is  a  pure  quad- 
ratic equation  ?    What  is  an  affected  quadratic  equation  ? 


198  RAY'S   ALGEBRA,    PART   FIRST. 

Art.  20§. — Every  equation  of  the  second  degree,  may  be  re* 
duced  to  one  of  the  forms  ax^=b,  or  ax^-\-bx=^c.  For,  in  an 
incomplete  equation,  all  the  terms  containing  x^  may  be  collected 
together,  and  then,  if  the  coefficient  of  x^  contains  more  than  one 
term,  it  may  be  assumed  equal  to  a  single  quantity,  as  a,  and  the 
sum  of  the  known  quantities,  to  another  quantity,  b,  and  then  the 
equation  becomes         ax^=b,  or  ax^ — 6=0. 

So  a  complete  equation  may  be  similarly  reduced ;  for  all  the 
terms  containing  x^  may  be  reduced  to  one  term,  as  ax"^;  and 
those  containing  x,  to  one,  as  bx;  and  the  known  terms  to  one,  as 
c;  then  the  equation  is  ax^-\-bx=c,  or  ax^-]-bx — c=0. 

Hence,  we  infer:  TJiat  every  equation  of  the  second  degree,  may 
he  reduced  to  an  incomplete  equation  involving  two  terms,  or  to  a  com- 
plete equation  involving  three  terms. 

Frequent  illustrations  of  these  principles  will  occur  hereafter. 

INCOMPLETE   EQUATIONS   OP   THE  SECOND   DEGREE. 

Art.  209. — 1.  Let  it  be  required  to  find  the  value  of  x  in  the 
equation  x^ — 16=0. 

Transposing,  x'^=:16 

Extracting  the  square  root  of  both  members, 

x  =±4,  that  is,  x-=-\-4:,  or  — 4. 
Verification.       (+4)2-10=16-16=0. 
or,  (-4)2-16=16-16=0. 

2.  Find  the  value  of  x  in  the  equation  5x^+4=49. 
Transposing,  5a;-=45 

Dividing,  x'^r=  9 

Extracting  the  square  root  of  both  sides, 
x=±3. 

2x^     3^.2 

3.  Find  the  value  of  x  in  the  equation  -^-~\ — j-=5f . 

o         4 

Clearing  of  fractions,  8x^+9x^=68 

Reducing,  17x^=68 

Dividing,  x'^-=  4 

Extracting  the  square  root,  x  =rd=2. 

4.  Given  ax^-\-b^^cx'^-{-d,  to  find  the  value  of  x. 

ax^ — cx^^^d — b 

or,  (a — c)3?^^d — b 

„     d-b 


-S- 


EQUATIONS  OF  THE  SECOND  DEGREE.  199 

From  the  preceding  examples,  we  derive  the 

RULE, 

FOR    THE     SOLUTION    OF    AN     INCOMPLETE    EQUATION    OF    THE    SECOND 
DEGREE. 

Reduce  the  equation  to  the  form  ax^=b.  Divide  both  sides  by  the 
coefficient  of  x\  and  then  extract  the  square  root  of  both  members. 

Art.  210. — If  we  take  the  equation  ax'^=b 

we  have  x^=^- 

a 

and  X  =rt* /- ;  that  is, 

If  we  assume  -=m^,  then  x'^=m'^ 
a 

By  transposing,  x"^ — m'^=Q 

By  separating  into  factors,     {x-\-m)[x — w)=0. 

Now,  this  equation  can  be  satisfied  in  two  ways,  and  in  two 
only;  that  is,  by  making  either  of  the  factors  equal  to  0. 

By  making  the  second  factor  equal  to  0,  we  have 
X — 7n=0,  or  a:=+wi. 

By  making  the  first  factor  equal  to  0,  we  have 
x-f-7;i=0,  or  x= — m. 

Since  the  equation  {x-[-m)[x — m)=:0,  can  be  satisfied  only  in 
these  two  ways,  it  follows,  that  the  values  of  x  obtained  from  these 
conditions,  are  the  only  values  of  the  unknown  quantity. 

Hence  we  conclude 

1st.  That  every  inco7nplete  equation  of  the  second  degree,  has  two 
roots,  and  only  two. 

2d.   That  these  roots  are  equal,  but  ham  contrary  signs. 
■    Find  the  roots  of  the  equation,  or  the  values  of  x,  in  each  of  the 
following  examples. 

1.  x'—S=2S Ans.  x=±6. 

2.  3x2— 15-=83+x2 Ans.  a:=±7. 

3.  aV— 62=0 Ans.  a;=±-. 

a 

4.  7x2—25=4x2—13 Ans.  x=±2. 

Review. — 208.  To  what  two  forms  may  every  equation  of  the  second 
degree  be  reduced  ?  Why  ?  209.  What  is  the  rule  for  the  solution  of  an 
incomplete  equation  of  the  second  degree  ?  210.  Show  that  every  incom- 
plete equation  of  the  second  degree,  has  two  roots,  and  only  two;  and  that 
those  roots  are  equal,  but  have  contrary  signs. 


200  RAY'S  ALGEBRA,   PART    FIRST. 

5.  5x2—2=8—35x2 \    .    Ans.  x-- 


4x 


6.  ix'-l=-^^+i Ans.x=±3. 


7.  ^-  +  12=^-+37| Ans.  x=±7. 


5x2  8x2 

^-  +  12-^ 

8.  (2x— 5)2=x2— 20x+73 Ans.  x=±4. 

9.  ax2 — b—{a~b)x^^c Ans.  x=±^/— t~  . 

-„    x+a  .  x—a      lOa- 

10. 1 — ^=-5 ; Ans.  x=±2a. 

X — a     x-f-a     x^ — a 

, ,     X — a     a — 2x    x2+5x  .  ,     ,— p- 

11. =-; — -7, Ans.  x=d=/a6. 

a         X — a      x^ — a^ 

QUESTIONS    PRODUCING   INCOMPLETE    EQUATIONS   OP  THE 
SECOND    DEGREE. 

Art.  211. — In  the  solution  of  a  problem  producing  an  equation 
containing  the  second  power  of  the  unknown  quantity,  the  equa- 
tion is  found  on  the  same  principle,  as  in  questions  producing 
equations  of  the  first  degree.     See  Art.  156. 

1.  Find  a  number,  whose  |  multiplied  by  its  f ,  will  be  equal 
to  60. 


2x.  .2x    4x 


Let  x=  the  number;  then  -^'X-—=z^—-:^(jO 
d      5      15 

4x2=900    - 

x2=225 
x=  15. 

2.  "What  number  is  that,  of  which  the  product  of  its  third  and 
fourth  parts  is  equal  to  108?  Ans.  36. 

3.  AVhat  number  is  that,  whose  square  diminished  by  16,  is 
equal  to  half  its  square  increased  by  16?  Ans.  8. 

4.  What  number  is  that,  whose  square  diminished  by  54,  is 
equal  to  the  square  of  its  half,  increased  by  54?  Ans.  12. 

5.  What  number  is  that,  which  being  divided  by  9,  gives  the 
same  quotient,  as  16  divided  by  the  number?  Ans.  12. 

6.  What  two  numbers  are  to  each  other  as  3  to  5,  and  the  dif- 
ference of  whose  squares  is  64  ? 

Let  3x;=^  the  less  number;  then  5x=  the  greater. 
And  (5xf-(3x)'-'=64 

Or  25x2— 9x''^=l  6x'^=64. 

From  which  x  =2;  hence,  3x=6  and  5x=10,  are 

the  numbers.     See  general  directions,  page  127. 

Keview. — 211.  In  the  solution  of  a  problem  producing  an  equation 
containing  the  second  power  of  the  unknoAvn  quantity,  upon  what  principle 
is  the  equation  found  ? 


EQUATIONS  OF  THE  SECOND  DEGREE.  201 

7.  What  two  numbers  are  those  which  are  to  each  other  as  3  to 
4,  and  the  difference  of  whose  squares  is  63  ?         Ans.  9  and  12. 

8.  What  two  numbers  are  those,  which  are  to  each  other  as  3 
to  4,  and  the  sum  of  whose  squares  is  100?  Ans.  6  and  8. 

9.  What  number  is  that,  to  which  if  3  be  added,  and  from  which 
if  3  be  subtracted,  the  product  of  the  sum  and  difference  is  40  ? 

Ans.  7. 

10.  The  breadth  of  a  lot  of  ground  is  to  its  length,  as  5  to  9, 
and  it  contains  1620  square  feet;  required  the  breadth  and  length. 

Ans.  Breadth  30,  length  54  feet. 

11.  A  man  purchased  a  farm,  giving  j^  as  many  dollars  per 
acre,  as  there  were  acres  in  the  farm ;  the  cost  of  the  farm  Avas 
1000  dollars;  required  the  number  of  acres  and  the  price  per 
acre.  Ans.  100  acres,  $10  per  acre. 

12.  What  two  numbers  are  those,  whose  sum  is  to  the  greater, 
as  10  to  7,  and  whose  sum,  multiplied  by  the  less,  produces  270? 

Ans.  21  and  9. 
Let  10a:=  their  sum;  then  7x=  the  greater,  and  3x=  the  less 
number. 

13.  What  two  numbers  are  those,  whose  difference  is  to  the 
greater  as  2  to  9,  and  the  difference  of  whose  squares  is  128? 

Ans.  18  and  14. 

14.  C  bought  a  number  of  oranges  for  48  cents,  and  the  price 
of  an  orange  was  to  the  number  bought,  as  1  to  3  ;  how  many  did 
he  buy,  and  how  much  a  piece  did  he  pay  ? 

Ans,  12  oranges,  at  4  cents  a  piece. 

15.  A  person  bought  a  piece  of  muslin  for  3  dollars  and  24 
cents,  and  the  number  of  cents  which  he  paid  for  a  yard,  was  to 
the  number  of  yards,  as  4  to  9 ;  how  many  yards  did  he  buy,  and 
what  was  the  price  per  yard?      Ans.  27  yds.,  at  12  cents  per  yd. 

16.  Find  two  numbers,  in  the  ratio  of  ^  to  |,  the  sum  of  whose 
squares  is  225.  Ans.  9  and  12. 

By  reducing  ?>  and  |  to  a  common  denominator,  we  find  they 
are  to  each  other  as  3  to  4.  Then  let  3a:  and  4x  represent  the 
numbers. 

17.  Find  three  numbers,  in  the  proportion  of  ^,  f ,  and  |,  the 
sum  of  whose  squares  is  724.  Ans.  12,  16,  and  18. 

18.  A  merchant  sold  a  piece  of  muslin  at  such  a  rate,  that  the 
price  of  a  yard  was  to  the  number  of  yards,  as  4  to  5  ;  but,  if  he 
had  received  45  cents  more  for  the  same  piece,  the  price  of  a  yard 
would  have  been  to  the  number  of  yards  as  5  to  4 ;  how  many 
yards  were  there  in  the  piece,  and  what  was  the  price  per  yard? 

Ans.  10  yards,  at  8  cents  per  yard. 


202  RAY'S   ALGEBRA,    PART    FIRST. 


COMPLETE   EQUATIONS   OF  THE   SECOiVD    DEGREE. 

1.  Let  it  be  required  to  find  the  values  of  x,  in  the  equation 

a;2_4a:+4=L 

It  is  evident,  from  Article   197,  that  the  first  member  of  this 
equation  is  a  perfect  square.     By  extracting  the  square  root  of 
both  members,  we  have    x — 2=d=l 
Whence  x=--2±l=2+l=3,  or  2-1=1. 

Verification.    (3)'^— 4X3+4=1,  that  is,  9-12+4=1 
also,  (1)2—4X1+4=1,  that  is,  1—  4+4=1. 

Hence,  x  has  two  values,  +3  and  +1,  either  of  which  verifies 
the  equation. 

2.  Let  it  be  required  to  find  the  value  of  x,  in  the  equation 

a;2+6x=16. 
If  the  left  member  of  this  equation  were  a  perfect  square,  we 
might  find  the  value  of  x,  by  extracting  the  square  root,  as  in  the 
preceding  example.  To  ascertain  what  is  necessary  to  be  added, 
to  render  the  first  member  a  perfect  square,  let  us  compare  it  with 
the  square  of  x+a,  which  is  x^-\-2ax-\-d^. 
We  find  x'=x' 

2ax  =^6x 
2a  =6 
«=3 

Hence,  by  adding  9,  wJiich  is  the  square  of  half  the  coefficient  of 
the  first  power  of  x,  to  each  member,  the  equation  becomes 

x2+6x+9=25 
Extracting  the  square  root,      x+3=±5 
Whence  x=— 3±5=+2,  or  —8. 

Either  of  which  values  of  x  will  verify  the  equation. 

Art.  212. — We  will  now  show  the  different  forms  to  which 
every  complete  equation  of  the  second  degree  may  be  reduced,  and 
illustrate  further,  the  principle  of  completing  the  square. 

Since  every  complete  equation  of  the  second  degree  may  be  re- 
duced to  the  form  ax^+6x=c,  if  we  divide  both  sides  by  a,  we  have 

,,6       c 

x^-\ — x=-. 

a      a 

.  .  h  c 

For  the  sake  of  simplicity,  let  -=2^,  and  -=^q.      The  equation 

then  becomes  x^-\-2px=^q        (1.) 

h  c        .  . 

If  -  is  negative,  and  -  positive,  the  equation  becomes 

a;2— 2_px=5        (2.) 


EQUATIONS  OF  THE  SECOND  DEGREE.  203 

b  .         .  .  c  . 

If  -  is  positive,  and  -  negative,  the  equation  becomes 

x'+2px=—q     (3.) 

.be 
Lastly,  if  -  and  -  are  both  negative,  the  equation  becomes 

a  Ctr 

x^—2px——q     (4.) 

Hence,  everi/  complete  equation  of  the  second  degree,  may  be  re- 
dticed  to  the  form  x^-^2px=q,  in  which  2p  and  q  may  be  either  pos- 
itive or  negative,  integral  or  fractional  quantities. 

"We  will  now  proceed  to  explain  the  principle,  by  which  the 
first  member  of  this  equation  may  always  be  made  a  perfect 
square. 

Since  the  square  of  a  binomial  is  equal  to  the  square  of  the  first 
term,  plus  twice  the  product  of  the  first  term  by  the  second,  plus 
the  square  of  the  second;  if  we  consider  x'^-{-2px  as  the  first  two 
terms  of  the  square  of  a  binomial,  x^  is  the  square  of  the  first  term 
{x),  and  2px,  the  double  product  of  the  first  term  by  the  second  ; 
therefore,  if  we  divide  2px  by  2x  (the  double  of  the  first  term),  or 
2p  by  2,  the  quotient,  p  {half  the  coefficient  of  x),  will  be  the  sec- 
ond term  of  the  binomial,  and  its  square,  p\  added  to  the  first 
member,  will  render  it  a  perfect  square.  But,  to  preserve  the 
equality,  we  must  add  the  same  quantity  to  both  sides.     This  gives 

x^-{-2pX'\-p^=^q-\-p'^ 
Extracting  the  square  root,      x-{-p  =zt\/q-\-p'^ 
Transposing,  x=^—pziz\/ q-tp^ 

It  is  obvious,  that  the  square  may  be  completed  in  each  of  the 
other  forms,  on  the  same  principle;  that  is,  by  taking  half  the 
coefficient  of  the  first  power  of  x,  squaring  it,  and  adding  it  to 
each  member.     Thus,  in  the  second  form     • 

x"^ — 2px=q 
x'- — 2px\p^=q^rV^ 

In  the  third  and  fourth  forms,  the  values  of  x  are  readily  ob- 
tained, in  the  same  manner. 

Collecting  the  four  difi'erent  forms  together,  and  the  values  of  x 
in  each,  we  have  the  following  table. 

(1.)     x^-^2ipx=q.  a-'=— yrhi/g+//. 

(2.)     x'—2yx=q.  x=-\-p±.Vq-\-f. 

(3.)     x'^2px=—q.  x=—pzt\/—q-\'P^. 

(4.)     .r^— 2jw=— 2-  x.=-\-pdt\/—q-{-p^. 


204  KAY'S   ALGEBRA,    PART   FIRST. 

Although  the  method  of  finding  the  value  of  x  is  the  same  in 
each  of  these  forms,  it  is  convenient  to  distinguish  between  them. 
See  Art.  215. 

From  the  preceding  we  derive  the 

RULE, 

FOR     THE     SOLUTION     OF     A     COMPLETE     EQUATION     OF     THE     SECOND 
DEGREE. 

1st.  Reduce  the  equation,  hy  clearing  of  fractions  and  transposi- 
tion [if  necessary),  to  the  form  ax^-^-hx^c. 

2d.  Divide  each  side  of  the  equation  hy  the  coefficient  of  x^,  and 
add  to  each  member  the  square  of  half  the  coefficient  of  the  first 
power  of  X. 

3d.  Extract  the  square  root  of  both  sides,  and  transpose  the  known 
term  to  the  second  member. 

EXAMPLES. 

1.  Find  the  roots  of  the  equation  x2+8x=33. 

Completing  the  square  by  taking  half  the  coefficient  of  a;(f ), 
squaring  it,  and  adding  the  square  to  each  member,  we  have 

x2+8a:-f  16^:33+16=49 
Extracting  the  root,  x+4=±7 

Transposing,  x= — 4zt-jf 

Whence  x=— 4+7=+3 

And  x=-4:-7=—U. 

Verif  cation.     (3)2+ 8(3)=33,  that  is,       9+24=33. 

Or         (-ll)2+8(-ll)=33,  that  is,  121-88=33. 

In  verifying  these  values  of  x,  it  is  to  be  noticed,  that  the  square 
of  — 11,  is  121,  and  that  8  multiplied  by  — 11,  gives  — 88. 

2.  Solve  the  equation  x^ — 6x=16. 
Completing  the  square, 

a;2-6a;+9=16+9=25 
Extracting  the  root,  x — 3=:±5 

Transposing,  aj=+3±5 

Whence  a:=+3+5=+8 

And  x=+3— 5=-2. 

Both  of  which  will  be  found  to  verify  the  equation. 

3.  Solve  the  equation  x2+6a;= — 5. 
Completing  the  square, 

x^+6x+9=9-5=4 
Extracting  the  root,  x+3=zb2 

Transposing,  a;:=— 3i±:2 

Whence  a:=-3+2=— 1 

And  x=— 3— 2=— 5. 


EQUATIONS  OF  THE  SECOND   DEGREE.  205 


4.  Find  the  values  of  «,  in  the  equation  x^ — 10x= — 24. 
Completing  the  square, 

x^—\  0x4-25=25—24=1 
Extracting  the  root,  a:— 5=d=l 

Transposing,  a:=5±l 

Whence  x=5+l=6 

And  aj=5 — 1=4. 

The  preceding  examples,  illustrate  the  four  different  forms, 
when  the  equation  is  already  reduced.  Equations  of  the  second 
degree,  however,  generally  occur  in  a  more  complicated  form,  and 
require  to  be  reduced  before  completing  the  square. 

5.  Find  the  values  of  x,  in  the  equation  Sx — 5= . 

Clearing  of  fractions.         Z:^ — 5x=7a:+36 
Transposing,  Sx^ — 12x=36 

Dividing,  x^ — 4x.=12 

Completing  the  square, 

x^— 4x+4=16 
Extracting  the  root,  x — 2=zb4 

Transposing,  x=-[-2±4 

Whence  x=:6,  or  — 2. 

12x''  13a; 

G.  Find  the  values  of  x,  in  the  equation  — ^ — |-x=52H — =-. 

Clearing  of  fractions,       12x'^+5x=260+13x 
Transposing  and  reducing, 

12x2— 8x=260 
Dividing,  x^ — |a;=^/. 

Here  the  coefficient  of  x  is  — f,  the  half  of  which  is — \  ;  tho 
Hquare  of  this  is  ^,  which  being  added  to  both  sides,  we  have 

^2      2^_l1 6  5i    1 19fi 

Extracting  the  root,  x— i=±  V 

^_  1  J_|_l_4 

Whence  x=4-5,  or— '/. 

EXAMPLES    FOB    PRACTICE. 

Note. — The  first  sixteen  of  the  following  Examples,  are  arranged  to 
illustrate  the  four  forms,  to  one  of  which  every  complete  equation  of  the 
second  degree  may  be  reduced. 

7.  x^-f  8x=20 Ans.  x=2,  or— 10. 

8.  x=^-M0x=80 Ans.  x=4,  or  — 20. 

9.  x2+7x=78 Ans.  x=6,  or  —13. 

10.  x*+3x=28 Ans.  x=4,  or  —7. 


*206  RAY'S   ALGEBRA,    PART   FIRST. 

11.  x2-10x=:24 Ans.  a:=12,  or  — 2. 

12.  a;2-8a;=20 Ans.  x^lO,  or  —2. 

13.  x^ — 5a:=6 Ans.  a;=6,  or — 1, 

14.  x-'— 21x=100 Ans.  a::==25.  or -4. 


15.  a:^-f6x= — 8 Ans.  a:= — 2,  or — 4. 

16.  a;^+4a;=— 3 Ans.  a;=— 1,  or  — 3 

17.  a;2+8x=— 15 Ans.  a;=— 3,  or  — 5. 

18.  x'^lx=—\2 Ans.  a:=-3,  or  — 4. 

19.  x'—Qx=-S Ans.  x-=4,  or  2. 

20.  a:^— 8x=— 15 Ans.  x=5,  or  3. 

21.  a;^— 10x=:-21 Ans.  a:=z7,  or  3. 

22.  x'^— 15x=— 54 Ans.  a:=9.  or  6. 


23.  3x'''-2a:+ 123=256 Ans.  x=7,  or  —  L^. 

24.  2x2-5a;=:12 Ans.  a;=i4,  or  — |. 

25.  2x=^+3x=65 Ans.  a:=5,  or  — '^^ 

2x^_5^ 
3  ~2 

27.  j^^=x-24 Ans.  x=60,  or  40. 

28.  x2— a;-40=:170 Ans.  x=15,  or  —14. 

29.  a:=— Ans.  x=2,  or  -3. 

X 

30.  x-l+-^=0 Ans.  a=3,or2. 

a;— 4 

31.  ^ ^=4 Ans.  a;=24,  or  ~6. 

4     X— 2 

45 


26.  -n T^— 3 Ans.  x=4:,  or  — \. 


32.  ia;'^-ix+|=8-fx-a:2+W-     •    •    •    Ans.  x=4,  or -y^. 

1      'iQ 

33.  9a;+-=— +4 Ans.  a;=2,  or  -  V . 

XX  ^ 

34.  x2+a;=30 Ans.  a:=5,  or  ~6. 

35.  ^*+-=^+^^ Ans.  x=2,  or  4. 

36.  2x^+92=3U- Ans.  x=4,  orllr]. 

37.  — x*^+x=o% Ans.  x=|,  or  |. 

38.  17x2-19x=30 Ans.  a;=2,  or— If. 

11  E  V  I  K  w. — 212.  To  what  form  may  every  complete  equation  of  the  sec- 
ond degree  be  reduced?  What  arc  the  four  forms  that  this  gives,  depend- 
ing on  the  signs  of  2p  and  q  ?  Explain  the  principle,  by  means  of  which  the 
first  member  of  the  equation  x'^-\-2p.v=:q  may  be  made  a  perfect  square. 
What  is  the  rule  for  the  solution  of  a  complete  equation  of  the  second  degree  ? 


EQUATIONS  OF  THE  SECOND  DEGREE.  207 


42. 


39.  3a:'^+5x=2 Ans.  x=::J,or-2. 

40.  4x-3x2=6x-8 _.    .  Ans.  a:=-^,  or -2. 

41.  x^— 4x=— 1.    .    .    .    Ans.  a:=2±>/3=3.732+,  or  .268— . 
4x     2x'     lOx     20 
ly         Q- — ~g ^ Ans.  a:== — 5,  or  f . 

.„    65x     10x2    13     2x  3 

~2         rF^^~n*'    *    ■     *  •    •    •     Alls.  x=354,  or  ^. 

"4-  ^=2^1 Ans..=12,or-2. 

X  7 

^^'  ^+60=3^5 Ans.x=.14,or-10. 

24 

46.  xH ^=3x— 4 Ans.  x=5,  or  -  2. 

X — 1 

^^    22-x     15— X  ,  ««        ,. 

47-  -o77-= p Ans.  x=36,  or  12. 

20        X — o 

x+3      7x      23  4^1 

48. ro=-T Ans.  x=4,  orl. 

X       x-fd      4 


50.  2ax— x2= — 2ab — b Ans.  x=2a+6,  or  —b. 

51.  x^ — 2ax=&^ — a^ Ans.  x=aH-6,  or  a — b. 

52.  x2+36x— 46'^-:^0 Ans.  x=-\-b,  or  —4b. 

53.  x"^ — ax — bx= — ab Ans.  x=H-a,  or  -{-b. 


54.  — ■ — = J Ans.  x=&d=i/«6H-^^ 

x-\-a      X — b 

55.  2bx'^-\-{a — 26)x=a Ans.  x=l,  or — ^. 

p^    x^       X     2a2  .  2a^        .      a^ 

ob.   -^  —  -==-—- Ans.  x=^—  and — r- 

a^       b      ¥  b  b 

57.  x'^ — («— l)x— a=0 Ans.  x=a,  or  — 1. 

58.  x"-^— (a+6— c)x— (aH-6)c Ans.  x=a+6,  or  — c. 

Art.  213.— The  Hindoo  method  of  solving  quadratics.— 
When  an  equation  is  brought  to  the  form  ax'^+/;x=c,  it  may  be 
reduced  to  a  simple  equation,  without  dividing  by  the  coefficient 
of  x^;  thus  avoiding  fractions. 

If  we  multiply  both  sides  of  the  equation  ax'^+6x=c,  by  a,  the 
coefficient  of  x*,  it  becomes  a?x^-{-abx=^ac. 

Now,  if  we  regard  d^x^-{-abx,  as  the  first  and  second  terms  of 
the  square  of  a  binomial,  db"^  must  be  the  square  of  the  first  term, 
and  abx  the  double  product  of  the  first  term  by  the  second.  Hence, 
the  first  term  of  the  binomial  is  i'' d^x'^^=ax ;  and  the  second  term, 
the  quotient  derived  from  dividing  abx  by  the  double  of  ax,  the 


208  RAY'S   ALGEBRA,    PART   FIRST. 

first  term ;  that  is,  ^ — =^.     Adding  the  square  of  ^  to  each  side, 

lidX  i4t  til 

the  equation  becomes  aV+«&x+-j-=ac+^. 

Now,  the  left  side  is  a  perfect  square ;  but  it  will  still  be  a  per- 
fect square,  if  we  multiply  both  sides  by  4,  which  will  clear  it  of 
fractions.     Thus,  4aV-|-4a6x+62=4ac4-62 
Extracting  the  square  root, 

2aa;+6=±|/4ac+6^ 

Whence  ^=W^^i«£±L'. 

Now,  it  is  evident,  that  the  equation  4aV+4a&x'+6"^=4ac-f  6^ 
may  be  derived  directly  from  the  equation  ax^-\-bx^^c,  by  multi- 
plying both  sides  by  4a,  the  coefficient  of  x^,  and  then  adding  to 
each  member,  the  square  of  6,  the  coefficient  of  the  first  power  of  x. 
This  gives  the  following 

RULE, 

FOR    THE     SOLUTION    OF    A     COMPLETE    EQUATION    OF    THE    SECOND 
DEGREE. 

Reduce  the  equation  to  the  form  ax'^-{-hx=c,  and  midtiply  both 
sides,  by  four  times  the  coefficient  ofx^.  Add  the  square  of  the  coef- 
ficient of  X  to  each  side,  and  then  extract  the  square  root.  This 
tcill  give  a  simple  equation,  from  which  x  is  easily  found. 

1 .  Given  3x^ — 5a:=28,  to  find  the  values  of  x. 
Multiplying  both  sides  by  12,  which  is  4  times  the  coefficient  of  x;^, 

36a;2-60x=336 
Adding  to  each  member  25,  the  square  of  5,  the  coefficient  of  x, 

36x2-60x+25=:361 
Extracting  the  root,  6a;—  5=±10 

6a;=5d=19=24,or-14 
a:=+4,  or  —J. 
By  the  same  rule,  find  the  values  of  the  unknown  quantity  in 
each  of  the  following  examples. 


2.  2x^+5x=33 Ans.  x=S,  or 

3.  5a:'^+2.T-=88 Ans.  x=4,  or 


1 1 

■    5    • 

4.  3x'^— x=70 Ans.  x=5,  or— V- 

5.  x^— x=42 Ans.  x=7,  or  —6. 

6.  ia;-^+^"-5=9| Ans.  x=6,  or -7|. 

If  further  exercises  are  desired,  the  examples  in  the  preceding 
article  may  be  solved  by  this  rule. 

K  E  v  I  E  w. — 213.  Explain  the  Hindoo  method  of  completing  the  square. 


EQUATIONS  OF  THE  SECOND  DEGREE.  209 


PROBLEMS  PRODUCIIV6   COMPLETE   EQUATIONS   OF   THE 
SECO]\D  DEGREE. 

Art.  214. — 1.  What  number  is  that,  whose  square,  diminished 
by  the  number  itself,  is  equal  to  20  ? 

Let  x=  the  number. 
Then  a;^— x=20 

Completing  the  square,  x^ — x+4-=20+?=V 
Extracting  the  root,  x — |=rfc| 

Whence  a;=+5,  or  — 4. 

Now  either  of  these  values  of  x  satisfies  the  equation  ;  but  the 
'negative  value  — 4,  does  not  fulfill  the  conditions  of  the  question 
in  an  arithmetical  sense.  But,  since  the  subtraction  of  a  negative 
quantity  is  equal  to  the  addition  of  a  positive  quantity,  the  ques- 
tion may  be  so  modified,  that  the  value  — 4,  will  be  a  correct 
answer  to  it,  the  4  being  considered  positive.  The  question  thus 
changed,  is:  What  number  is  that,  whose  square  increased  by  the 
number  itself,  is  equal  to  20? 

2.  A  person  buys  several  oranges  for  60  cents ;  had  he  bought 
3  more  for  the  same  sum,  each  orange  would  have  cost  him  1  cent 
less  ;  how  many  did  he  buy? 

Let  a:=  the  number  he  bought. 

Then  — =  the  price  of  each  one. 

X 

And  — ro=  the  price  of  one,  had  he  bought  3  more  for  60  cents. 

X-\-u 

m,       .  60       60       , 

Therefore,  ro=l 

X      x-^3 

Clearing  of  fractions,  and  reducing, 

a:2-f3a:=180 

Completing  the  square,         a;2+3x+|=:f +  I80=^f  ^. 

Extracting  the  root,  a:+|=±V 

Whence  a:=+12,  or  —15. 

Now  either  of  these  values,  taken  with  its  proper  sign,  satisfies 
the  equation  from  which  it  was  derived ;  but  the  value  12  is  the 
only  one  that  satisfies  the  conditions  of  the  question. 

Since  -f  f  =— 4  and  -^^=^^=—5;  and  since  buying  and  selling 
are  opposite  operations,  the  result,  — 15,  is  the  answer  to  this 
question.  A  person  sells  several  oranges  for  60  cents.  Had  he 
sold  3  less  for  the  same  sum,  he  would  have  received  one  cent  more 
for  each.     How  many  oranges  did  he  sell  ? 

R  E  M  A  R  K.— From  the  two  preceding  examples,  we  see,  that  the  root 
which  is  obtained,  from  giving  the  plus  sign  to  the  radical,  satisfies  both 

18 


210  RAY'S    ALGEBRA,    PART   FIRST. 

the  conditions  of  the  question,  and  the  equation  derived  from  it;  while  the 
other  root  satisfies  the  equation  only. 

We  see,  also,  that  the  root  which  arises  from  giving  the  radical  the  nega- 
tive sign,  may  be  regarded  as  the  answer  to  a  question  differing  from  the 
one  proposed  in  this ;  that  certain  quantities  which  were  additive,  have 
become  subtracfive,  and  reciprocally. 

Sometimes,  however,  as  in  the  following  example,  both  values  of  the 
unknown  quantity  satisfy  the  conditions  of  the  question. 

3.  Find  a  number,  whose  square  increased  by  15,  shall  be  8 
times  the  number. 

Let  x=  the  number;  then  x^-{-l5=^8x 
Or  x-'-8x=^~l5 

Whence  x=6,  or  3. 

Either  of  which  fulfills  the  conditions  of  the  question. 

"When  there  are  two  unknown  quantities  in  a  problem,  that  can 
be  solved  by  the  use  of  one  symbol,  the  two  values  of  the  symbol 
generally  give  both  values  of  the  unknown  quantity,  as  in  the 
following  question. 

4.  Divide  the  number  24  into  two  such  parts,  that  their  produet 
fc;hall  be  95. 

Let  x=  one  of  the  parts  ;  then  24 — z=  the  other. 
And  x{24—x)=d5 

Or  x2— 24x=— 95 

Whence  a:=^19  and  5 

And  24—x=5,  or  19. 

5.  There  are  three  numbers,  such  that  the  product  of  the  first 
and  third  is  equal  to  the  square  of  the  second  ;  the  sum  of  the 
first  and  second  is  10,  and  the  third  exceeds  the  second,  by  24; 
required  the  numbers. 

Let  a;==  the  first;  then  10 — x=  the  second, 
And  10— x+24i=34— x=  the  third. 
Also  {10—xY^x{M—x) 

Or  100-20a;+a;=^=34x— x"^ 

From  which,  a:=25,  or  2. 

When  a::=:25,  10— a:=— 15,  34 — x=9,  and  the  numbers  are  25, 
— 15,  and  9. 

When  x=2,  10— x=8,  34— a;=32,  and  the  numbers  are  2,  8, 
and32. 

Both  these  sets  of  values  satisfy  the  question  in  an  algebraic 
sense ;  only  the  last,  however,  satisfies  it  in  an  arithmetical  sense. 
Let  us  endeavor  to  ascertain  how  the  question  must  be  modified, 
80  that  the  first  set  of  numbers  shall  satisfy  it  in  an  arithmetical 
sense.  . 


EQUATIONS  OF  THE  SECOND  DEGREE.  211 

The  meaning  of  the  negative  solution  — 15,  will  be  understood 
by  considering  that  the  addition  of  a  negative  quantity,  is  the  same 
us  the  subtraction  of  the  same  quantity  taken  positively  (Art  61). 
The  first  condition  of  the  question  then  becomes  25+( — 15)=25 
_(4-15)=25— 15==10;  and  the  second  is  9-(— 15)==9+(+15) 
—9+15=24.  This  indicates,  that  —-15  may  be  changed  to  +15, 
provided,  that  instead  of  the  condition  of  the  sum  of  the  first  and 
second  numbers  being  10,  their  dijference  be  10;  and  the  second 
condition  may  for  a  similar  reason,  be  changed  into  this,  that  the 
sum  of  the  second  and  third  is  24.  The  question,  M'ith  these  modi- 
fications, M^ould  be:  AVhat  three  numbers  are  those,  such  that  the 
product  of  the  first  and  third,  is  equal  to  the  square  of  the  second; 
the  difference  of  the  first  and  second  is  10;  and  the  sum  of  the 
second  and  third  is  24  ? 

Remark. — In  the  following  examples,  the  pupil  is  required  to  find 
only  that  value  of  the  unknown  quantity,  which  satisfies  the  conditions  of 
the  question  in  an  arithmetical  sense.  It  forms,  however,  a  good  exercise 
for  advanced  pupils,  to  determine  the  negative  value,  and  then  to  modify 
the  question,  so  that  this  value  shall  satisfy  the  conditions  in  an  arithmetical 
sense. 

6.  Find  a  number,  such  that  if  its  square  be  diminished  by  6 
times  the  number  itself,  the  remainder  shall  be  7.  Ans.  7. 

7.  Find  a  number,  such  that  if  its  square  be  increased  by  8 
times  the  number  itself,  the  sum  shall  be  9.  Ans.  1. 

8.  Find  a  number,  such  that  twice  its  square,  plus  3  times  the 
number  itself,  shall  be  65.  Ans.  5. 

9.  Find  a  number,  such  that  if  its  square  be  diminished  by  1 , 
and  I  of  the  remainder  be  taken,  the  result  shall  be  equal  to  5 
times  the  number  divided  by  2.  Ans.  4. 

10.  Find  a  number,  such  that  if  44  be  divided  by  the  number 
diminished  by  2,  the  quotient  shall  be  equal  to  \  of  the  number, 
diminished  by  4.  Ans.  24. 

1 1 .  Find  two  numbers,  whose  difference  is  8,  and  product  240. 

Ans.  12  and  20. 

12.  A  person  bought  a  number  of  sheep,  for  80  dollars  ;  if  he 
had  bought  4  more  for  the  same  money,  he  would  have  paid  1 
dollar  less  for  each;  how  many  did  he  buy?  Ans.  16. 

13.  There  are  two  numbers,  whose  difference  is  10,  and  if  600 
bo  divided  by  each,  the  difference  of  the  quotients  is  also  10; 
what  are  the  numbers  ?  Ans.  20  and  30. 

14.  A  pedestrian,  having  to  walk  45  miles,  finds  that  if  he  in-  , 
creases  his  speed  \  a  mile  an  hour,  he  wnll  perform  his  task  l\ 


212  RAY'S   ALGEBRA,    PART    FIRST. 

hours  sooner,  than  if  he  walked  at  his  usual  rate;  what  is  that 
rate  ?  Ans.  4  miles  per  hour. 

15.  Divide  the  number  14  into  two  parts,  the  sum  of  whose 
squares  shall  be  100.  Ans.  8  and  6. 

16.  In  an  orchard  containing  204  trees,  there  are  5  more  trees 
in  a  row  than  there  are  rows;  required  the  number  of  rows,  and 
the  number  of  trees  in  a  row.       A.  12  rows,  and  17  trees  in  a  row. 

17.  A  schoolboy,  being  asked  the  ages  of  his  sister  and  himself, 
replied,  that  he  was  4  years  older  than  his  sister,  and  that  twice 
the  square  of  her  age,  was  7  less  than  the  square  of  his  own ; 
required  their  ages.  Ans.  13  and  9  yrs. 

18.  A  and  B  start  at  the  same  time  to  travel  150  miles;  A 
travels  3  miles  an  hour  faster  than  B,  and  finishes  his  journey  83- 
hours  before  him  ;  at  what  rate  per  hour  did  each  travel? 

Ans.  9  and  6  miles  per  hour. 

19.  A  company  at  a  tavern  had  1  dollar  and  75  cents  to  pay; 
but  before  the  bill  was  paid  two  of  them  went  away,  when  those 
who  remained  had  each  10  cents  more  to  pay;  how  many  were  in 
the  company  at  first?  Ans.  7. 

20.  The  product  of  two  numbers  is  100,  and  if  1  be  taken  from 
the  greater,  and  added  to  the  less,  the  product  of  the  resulting 
numbers  is  120;  what  are  the  numbers?  Ans.  25  and  4. 

Let  x=  the  larger  number;  then  - —  =  the  smaller. 

21.  If  4  be  subtracted  from  a  father's  age,  the  remainder  will' 
be  thrice  the  age  of  the  son;  and  if  1  be  taken  from  the  son's  age, 
lialf  the  remainder  will  be  the  square  root  of  the  father's  age. 
Required  the  age  of  each.  Ans.  49  and  15  3a's. 

Let  x^=  the  father's  age;  then  — ^ — =  the  son's  age. 

o 

22.  A  young  lady  being  asked  her  age,  answered,  "  If  you  add 
the  square  root  of  my  age  to  |  of  my  age,  the  sum  will  be  10." 
Required  her  age.  Ans.  16  yrs. 

23.  What  number  is  that,  from  which,  if  7I  of  its  square  root 
be  taken,  the  remainder  will  be  22  ?  Ans.  25. 

24.  A  merchant  bought  a  piece  of  muslin  for  6  dollars;  after 
cutting  off  15  yards,  he  sold  the  remainder  for  5  dollars  40  cents, 
l)y  which  he  gained  1  cent  a  yard  on  the  amount  sold  ;  how  many 
vards  did  he  buy,  and  at  Avhat  price? 

Ans.  75  yds.,  at  8  cts.  per  yd. 

25.  A  man  bought  a  horse,  which  he  afterward  sold  for  24  dol- 
lars, and  by  so  doing,  lost  as  much  per  cent,  upon  the  price  of  his 
purchase,  as  the  horse  cost  him  ;  what  did  he  pay  for  the  horse  ? 

Ans.  $60,  or  $40. 


EQUATIONS  OF  THE  SECOND  DEGREE.  213 

PROPERTIES    OP    THE  ROOTS   OF   A  COMPLETE    EQUATION 
OF  THE    SECOIVD   DEGREE. 

NoTK  TO  Teachers. — This  subject  may  be  omitted  entirely,  by 
the  younger  class  of  pupils ;  and  passed  over,  by  those  more  advanced, 
until  the  book  is  reviewed. 

Art.  215. — The  pupil  may  have  learned  already,  by  inference, 
from  the  solution  of  the  preceding  examples,  that  an  equation  of 
the  second  degree  has  two  roots,  that  is,  that  the  unknown  quan- 
tity has  two  values.  This  principle  may  be  proved  directly,  as 
follows. 

The  general  form  to  which  every  complete  equation  of  the  sec- 
ond degree  may  be  reduced,  is  x^-\-2px=q ;  in  which  2p  and  q 
may  be  either  both  positive  or  both  negative,  or  one  positive  and 
the  other  negative.     Completing  the  square,  we  have 
x'+2px-{-p'=q+p' 

Now,  the  first  member  is  equal  to  {x-j-pY,  and  if,  for  the  sake  of 
simplicity,  we  assume  q-\-j)'^=ni'^,  that  is,  y/q-\-p^=m,  then 

Transposing,  {^'hpY — wi'^=0. 

But,  since  the  left  hand  member  of  this  equation,  is  the  differ- 
ence of  two  squares,  it  may  be  resolved  into  two  factors.  Art.  94. 
This  gives  {x'\-p-\-m){x-\-p — wi)=0 

Now,  this  equation  can  be  satisfied  in  two  ways,  and  in  oiily  two; 
that  is,  by  making  either  of  the  factors  equal  to  0. 

If  we  make  the  second  factor  equal  to  zero,  we  have 
x-^p — m=0 
Or,  by  transposing,  x= — p-{-m^= — p^V^.'^P^ 

If  we  make  the  first  factor  equal  to  zero,  we  have 

Or,  by  transposing,  x=— jp — m=—p — i/^+p^ 

Hence,  we  conclude, 

1  St.  TJiat  every  equation  of  the  second  degree,  lias  tivo  roots,  and 
only  two. 

2d.  That  every  complete  equation  of  the  second  degree,  reduced  to 
the  form  x^-{-2px=q,  may  be  decomposed  into  two  binomial  factors, 
of  which  the  first  term  in  each  is  x,  and  the  second,  the  two  roots  with 
their  signs  changed. 

Thus,  the  two  roots  of  the  equation  x^— 5a:=— 6,  or  x' — 5x4-6 
-=0,  are  x=2  and  x=3;  hence,  x"^— 5x+6=(x— 2){x — 3). 

From  this,  it  is  evident,  that  the  direct  method  of  resolvmg  a 
cjuadratic  trinomial  into  its  factors,  is  to  place  it  equal  to  zero,  and 
then  find  the  roots  of  the  equation.  In  this  manner,  let  the 
learner  solve  the  questions  on  page  72. 


214  RAY'S   ALGEBRA,   PART   FIRST. 

By  reversing  the  operation,  we  can  readily  form  an  equation, 
whose  roots  shall  have  any  given  values. 

Thus,  let  it  be  required  to  form  an  equation  whose  roots  shall 
be  4  and  — 6. 

We  must  have  a:=    4  or  x — 4=0 

And  x= — 6  or  x+6=0 

Hence,  (x— 4)(x+6)=a:2+2a;— 24=0 

Or  a:2+2x=24. 

Which  is  an  equation  whose  roots  are  +4  and  — 6. 

1.  Find  an  equation  whose  roots  are  7  and  10. 

Ans.  a;-^-17a:=— 70. 

2.  Find  an  equation  whose  roots  are  — 3  and  — 1. 

Ans.  x^-\-4x= — 3. 

3.  Find  an  equation  whose  roots  are  +2,  and  — 1. 

Ans.  x"^ — x=2. 

Art.  216. — Resuming  the  equation  x^-\-2px=q. 
The  first  value  of  X  is         — p+i/^+/>^ 
The  second  value  of  x  is    — p — \/9'\~P^ 

Their  sum  is  — 2p,  which  is  the  coefficient  of 

X,  taken  with  a  contrary  sign.     Hence,  we  conclude, 

Thai  the  sum  of  the  roots  of  an  equation  of  the  second  degree,  re- 
duced to  the  form  x'^-\-2px^=q,  is  equal  to  the  coefficient  of  the  first 
power  of  X  taken  with  a  contrary  sign. 

If  we  take  the  product  of  the  roots,  we  have 
First  root=:  —p-\-V Q.^P^ 

Second  root=  — jP— l/^+i^^ 

f—pVq+p' 

■\-pV  q-{-p''—[q+p^) 
f   ,    .    .    .    ~{q^p')=-q. 
But  — q  is  the  known  term  of  the  equation,  taken  with  a  con- 
trary sign.     Hence,  we  conclude, 

That  the  product  of  the  two  roots  of  an  equation  of  the  second  de- 
gree, reduced  to  the  form  x'^-{-2px=q,  is  equal  to  the  known  term 
taken  with  a  contrary  sign. 

Remark  . — In  the  preceding  demonstrations,  we  have  regarded  2p  and 
q  as  both  positive  ;  the  same  course  of  reasoning,  however,  will  apply  when 
they  are  both  negative,  or  when  one  is  positive  and  the  other  negative ;  so 
that  the  conclusions  are  true  in  each  of  the  four  different  forms. 

Art.  SIT".— In  the  equation  x'^-\-2px^^q,  or  first  form, 
the  two  values  of  x  are         —p-\-V^'^P^ 
And  —p — \/'q-\-p^. 


EQUATIONS  OF  THE  SECOND   DEGREE.  215 


If  we  examine  the  part  \/q-\-p\  we  see  that  its  value  must  be 
a  quantity  greater  than  p,  since  the  square  root  of  p^  alone,  is  p. 
Hence,  the  first  root  is  the  difference  between  p  and  a  positive 
quantity  greater  than  p ;  therefore,  it  is  essentiallij  positive. 

If  we  take  the  negative  value  of  the  radical  part,  the  second 
root  is  equal  to  the  sum  of  two  negative  quantities,  one  of  which 
is  p,  and  the  other  a  quantity  greater  thanj?;  therefore,  it  is 
essentially  negative.  Since  the  first  root  is  the  difference,  and  the 
second  root  the  sum,  of  the  same  two  quantities,  the  second,  or 
negative  root,  is  necessarily  greater  than  the  first,  or  positive  root. 
See  questions  7,  8,  9,  10,  page  205^ 

In  the  equation  x^ — 2px=q,  or  second  form, 
the  two  values  of  x  are        -Vp-\-V^-\-P^ 
And  -{-p — Vq-\-pK 

The  quantity  under  the  radical  being  the  same  as  in  the  pre- 
ceding form,  its  square  root  is  greater  than  p.  The  first  root  then, 
is  the  sum  of  two  positive  quantities,  one  of  which  is  p,  and  the 
other  a  quantity  greater  than^;  therefore,  it  is  essentially  positive. 

If  we  take  the  negative  value  of  the  radical  part,  the  second 
root  is  equal  to  the  difference  between  p,  and  a  negative  quantity 
greater  than  p ;  therefore  it  is  essentially  negative. 

Since  the  first  root  is  the  sum,  and  the  second  root  the  difference 
of  the  same  two  quantities,  the  first,  or  positive  root,  is  greater 
than  the  second,  or  negative  root.  See  questions  11,  12,  13,  14, 
page  306. 

In  the  equation  x'^-\-2px=^ — q,  or  third  form, 
the  two  values  of  x  are     —p-\-V — 5'+i?^ 
And  —p — i/ — q-\-p^. 

If  we  examine  the  radical  part,  \/ — 5'+i?^  we  see,  that  its  value 
must  be  a  quantity  less  than  p,  since  the  square  root  of  j^'-*  without 
its  being  diminished,  isp;  hence,  the  first  root  is  the  difference 
between  — p,  and  a  positive  quantity  less  than  p;  therefore,  it  is 
essentially  negative. 

If  we  take  the  negative  value  of  the  radical  part,  the  second 
root  is  equal  to  the  sum  of  two  negative  quantities;  therefore,  it 
is  essentially  negative. 

Review. — 215.  To  what  general  form,  may  every  equation  of  the  sec- 
ond degree,  containing  one  unknown  quantity,  be  reduced  ?  Show  that 
every  equation  of  the  second  degree  has  two  roots,  and  only  two.  216.  To 
what  is  the  sum  of  the  roots  of  an  equation  of  the  second  degree  equal? 
To  what  is  the  product  equal?  217.  Show  that  in  the  first  form  one  of  tho 
roots  is  positive,  and  the  other  negative ;  and  that  tho  negative  root  is 
greater  than  the  positive. 


216  RAT'S   ALGEBRA,    PART    FIRST. 

Hence,  in  the  third  form,  both  roots  are  negative.  See  ques- 
tions 15,  16,  17,  18,  page  206. 

In  the  equation  x^ — 2px= — q,  or  fourth  form,  the  two  values  of 
X  are  +i?+l/— 9+i>^ 

And  -\-p — i/ — q-\-p^. 

The  value  of  the  radical  part,  being  the  same  as  in  the  pre- 
ceding form,  it  is  less  than  p.  The  first  root,  then,  is  the  sum  of 
two  positive  quantities,  therefore,  it  is  essentially  positive. 

The  second  root  is  the  difference  between^,  and  a  negative 
quantity  less  than  p,  therefore,  it  is  essentially  positive. 

Hence,  in  the  fourth  form^  both  roots  are  positive.  See  ques- 
tions 19,  20,  21,  22,  page  206. 

Art.  2 is* — In  the  third  and  fourth  forms,  the  radical  part  is 
|/ — q-\-p^'  Now,  if  q  is  greater  than  p'^,  this  is  essentially  nega- 
tive, and  we  are  required  to  extract  the  square  root  of  a  negative 
quantity,  which  is  impossible.  See  Art.  195.  Therefore,  in  the 
third  and  fourth  forms,  when  q  is  greater  than  p'^,  that  is,  when 
the  known  term  is  negative,  and  greater  than  the  square  of  half 
the  coefficient  of  the  first  power  of  x,  both  values  of  the  unknown 
quantity  are  impossible.     What  is  the  cause  of  this  impossibility? 

To  explain  this,  we  must  inquire  into  what  two  parts,  a  num- 
ber must  be  divided,  so  that  the  product  of  the  parts  shall  be  the 
greatest  possible. 

Let  2p  represent  any  number,  and  let  the  parts,  into  which  it  is 
supposed  to  be  divided,  be  2)-\-z  and  p — z.  The  product  of  these 
parts  is  {p~\'^)[p — z)=p'^ — z^. 

Now,  this  product  is  evidently  the  greatest,  when  z-  is  the  least; 
that  is,  when  z"^  or  z  is  0.  But,  when  z  is  0,  the  parts  are  ^  and 
p,  that  is,  when  a  number  is  divided  into  iivo  equal  parts,  their  pro- 
duct is  greater  than  that  of  any  other  tivo  parts  into  which  the  num- 
ber can  be  divided.  Or,  as  the  same  principle  may  be  otherwise 
expressed,  the  product  of  any  two  unequal  numbers  is  less  than  the 
square  of  half  their  sum. 

As  an  illustration  of  this  principle,  take  the  number  10,  and 
divide  it  into  different  parts. 

10=9+1,  and  9X1=  9 
10=8+2,  and  8X2=16 
10=7+3,  and  7X3=21 
10=6+4,  and  6X4=24 
10=5+5,  and  5X5=25 

Review. — 217.  Show  that  in  the  second  form,  one  root  is  positive  and  the 
other  negative ;  and  that  the  positive  root  is  greater  than  the  negative. 
Show  that  in  the  third  form,  both  roots  are  negative.  Show  that  in  the 
fourth  form,  both  roots  are  positive. 


EQUATIONS  OF  THE  SECOND  DEGREE.  217 


We  thus  see,  that  the  product  of  the  parts  becomes  greater  as 
they  approach  to  equality,  and  that  it  is  the  greatest  when  they 
are  equal  to  each  other. 

Now,  in  Art.  215,  it  has  been  shown,  that  2p,  the  coefficient  of 
the  first  power  of  x,  is  equal  to  the  sum  of  the  two  values  of  x, 
and  that  q  is  equal  to  their  product.  But,  when  q  is  greater  than 
^^  we  have  the  product  of  two  numbers,  greater  than  the  square 
of  half  their  sum,  which,  by  the  preceding  theorem,  is  impossible. 
If,  then,  any  problem  furnishes  an  equation  in  which  the  known 
term  is  negative,  and  greater  than  the  square  of  half  the  coeffi- 
cient of  the  first  power  of  the  unknown  quantity,  we  infer,  that 
the  conditions  of  the  problem  are  incompatible  with  each  other. 
The  following  is  an  example. 

Let  it  be  required  to  divide  the  number  12  into  two  such  parts, 
that  their  product  shall  be  40. 

Let  X  and  12 — x  represent  the  parts. 
Then  a:(12— a:)=:40,  or  a:'^— 12a;=— 40 

x'—l2x-{-S6^-4: _ 

a;— 6=dbi/— 4,  and  a;=6±v/— 4. 

These  expressions  for  the  values  of  x,  show  that  the  problem  is 
impossible.  This  we  also  know,  from  the  preceding  theorem, 
since  the  number  12  can  not  be  divided  into  any  two  parts,  whose 
product  will  be  greater  than  36 ;  thus,  the  algebraic  solution  ren- 
ders manifest  the  absurdity  of  an  impossible  problem. 

Remarks  . — 1st.  When  the  coeflScient  of  x^  is  negative,  as  in  the  equa- 
tion — x^-{-mu:=u,  the  pupil  may  not  perceive  that  it  is  embraced  in  the  four 
general  forms.  This  difficulty  is  obviated,  by  multiplying  both  sides  of  tho 
equation  by  — 1. 

2d.  Since  the  sign  of  the  square  root  of  x^,  or  of  ix-\-p)^,  is  :±:,  it  might 
seem,  that  when  x^=^m^,  we  should  have  -\-x=^-\-m,  that  is,  -\^x=-\-m, 
and  — x=-\-m ;  such  is  really  the  case,  but — x=-\-vi,  is  the  same  as 
-}-«== — m,  and  — x= — ni,  is  the  same  as  -\-x=-l-m.  Hence,  -\-x=-\-m, 
embraces  all  the  values  of  x.  In  the  same  manner,  it  is  necessary  to  take 
only  the  plus  sign  of  the  square  root  of  {x-\-p)'^. 

EQUATIONS   OP  THE  SECOIVD    DEGREE,    COIVTAIWING  TWO 
UARNOWM   QUAIVTITIES. 

Note. — A  full  discussion  of  equations  of  this  class  does  not  properly 
belong  to  an  elementary  treatise.  Indeed,  no  directions  can  be  given,  that 
will  be  applicable  to  all  cases.  The  general  method  of  treating  the  sub- 
ject, consists  in  presenting  the  solution  of  a  variety  of  examples,  and  then 
furnishing  others  for  the  exercise  of  the  student.  The  following  examples 
are  intended  to  embrace  only  those  capable  of  solution  by  simple  methods. 
Bee  Ray's  Algebra,  Part  Second. 

19 


218  RAY'S   ALGEBRA,    PART   FIRST. 


Art.  219. — In  solving  equations  of  the  second  degree,  contain- 
ing two  unknown  quantities,  the  first  step  is  to  eliminate  one  of 
them,  so  as  to  obtain  a  single  equation  involving  only  one  unknown 
quantity.  The  elimination  may  be  performed  by  either  of  the 
three  methods  already  given.  See  Articles  158,  159,  160.  When 
a  single  equation  is  thus  obtained,  the  value  of  the  unknown  quauv 
tity  is  to  be  found  by  the  rules  already  given. 

KX.  4IV1PLKS. 

1.  Given  x  -y — 2  and  x'-^y'—YOO,  to  find  x  and  y. 
By  the  first  equation,  x—y-\-2 
Substituting  this  value  of  x,  in  the  second, 

(y+2)-^+y^=100 
From  which  we  readily  find,   y=6,  or  — 8 
Hence,  '      a:=?/4-2=8,  or  — G. 

2.  Given  x-\-y=^S,  and  x?/=15,  to  find  x  and  y. 
From  the  first  equation,  x=8 — y 
Substituting  this  value  of  x,  in  the  second, 

y{S-y)=Vo 
Or  y2-8?/=-15 

From  which  y  is  found  to  be  5  or  3. 
Hence,  x=^3,  or  5. 

There  is  a  general  method  of  solving  questions  of  this  form, 
without  completing  the  square,  with  which  pupils  should  be  ac- 
quainted.    To  explain  it,  suppose  we  have  the  equations 

x^tj=a 

xy=b 

Squaring  the  first,        x^-i-2xy-i-y'^=a^ 

Multiplying  the  second  by  4,  4xy=4b 

Subtracting,  x^—2xy-\-y^=^^ — 46 

Extracting  the  square  root,      x — ?/=±]/a^ — 46 
But  x-\-y=a 

Adding  2a;— adry^a"^— 46 

Or  a;=:Jia±Ji/«'— 46 

Subtracting,  2y=a=^\/ d^—4b 

Or  y=:^a=^iy  a'—^b. 

If  we  have  the  equations  x — y=a  and  xy=b,  we  may  find  the 
values  of  x  and  y,  in  a  similar  manner,  by  squaring  each  member 
of  the  first  equation,  and  adding  to  each  side  4  times  the  second. 
Then,  extracting  the  square  root,  we  obtain  the  value  of  x-\-y 
=±v/a'^-f  46;  from  which,  andic— y=:a,  we  find  x=^aziz^V d^-\-4b, 
and  yr-^  Jaq=^i/aH^6. 


EQUATIONS  OF  THE  SECOND  DEGREE.  219 

3.  Given  x-\-i/=a  and  x^-}-t/^=b,  to  find  x  and  y. 
Squaring  the  first,         x^-\-2x7/-\-7/^=a'^  (3.) 

But,  x'+y'=b  (2.) 

Subtracting,  2x7/=a^ — b       (4.) 

Take  (4)  from  (2),        x^—2xi/+f=2b—a'' 
Extracting  the  root  x — ^y=±v/26 — d^ 

x-[-y=a 


Adding  and  dividing,  x=larfclv^26— a' 

Subtracting  and  dividing,  yz=^arpj^l/26 — a^. 

4.  Given  x^-]-i/=a  and  xy:=b,  to  find  x  and  y. 
Adding  twice  the  second  to  the  first, 

.    x'-\-2xij-^if=a^2b 
Extracting  the  square  root,        a;+y=±i/a+26 
Subtracting  twice  the  second  from  the  first, 
x"^ — 2xy-{-y'^=:a — 26 
Extracting  the  square  root,        x — y=d=>/a — 26 
Whence  x=dz^  l/'a+26±  1  /a— 26 


And  y=±ii/a+26=Fi>/a-26. 

5.  Given  x^-\-y^=a  and  x-^y=b,  to  find  x  and  ?/. 
Dividing  the  first  by  the  second, 

x^—xj/+t/=-^        (3.) 

Squaring  the  second,    x'^+2xy^y^=^¥       (4.) 

Subtracting  (3)  from  (4),  3xy=— r- 

^  6' — a  -_  . 

Or  ^^"""36"  ^   ^^ 

4a.— 6' 
Take  (5)  from  (3),       x'-2xy+y'=:—^^ 

.        ,  .     /  /  4«-6'  ^ 

Extracting  the  root,  x—y—-z±zyl  I  — qr —  1 

But  «+y=6 

,,      ,    l/4a-b'  \ 
Whence  a:=A6±2W  [      3^      j 

And  y=J6T-l^(li^). 

In  a  similar  manner,  if  we  have  x^—if~-a  and  x—y=b  we  find 


220  RAY'S   ALGEBRA,    PART   FIRST. 


EXAMPLES. 

6.  x'+f=34  I Ans.  x=it5. 

x'-f=l6S 2/=d=3. 

7.  x+y=^lQ)      Ans.  a:i=9,  or  7. 

X7/=QS  )  i/=7,  or  9. 

.  8.  X — ?/=5    I  Ans.  x=^9,  or  —4. 

x7/=36  }  y=4,  or  — 9. 

9.  X  -\-y  =Q    ] Ans.  x=7,  or  2. 

a;2+y=53J y^2,ovl. 

10.  x-y=5    I Ans.  x=8,  or  — 3. 

x2+/=73  I y=S,  or  -8. 

11.  7?-^7/=\^2)      Ans.  x=5,  or3. 

X  +?/  =8      J y— 3,  or  5. 

12.  x3-y3=:208  I      Ans.  x=6,  or  —2. 

a;  —ij  -=4      i      2/=2,  or  — 6. 

13.  x'+y=19(a;+y)  I Ans.  x=5,  or  -3. 

a;  — y  =3  i y=2,  or  —5. 

14.  ic+y=ll  1 Ans.  a:=0. 

x"^ — y^=ll  J Z/=S- 

15.  (x-3)(y+2)-=12| Ans.  a:=6,  or —3. 

xy=\2  i y=2,  or  —4. 

16.  y — a:=:2  )        Ans.  x=^2,  or  — 3. 

3a;?/=10x+// )  y— 4,  or    If. 

17.  3a;*''+2a:?/=24|  Ans.  a:=2,  or  — f  « 

bx  — 3?/  =1     J  y=3,  or  — 

1  ,  1 


1  99 
5?  ■ 


18.  -+-=§ 

1     ,     1_13 


Ans.  a:=2,  or  3. 


y=3,  or  2. 


19.  a: — y=2  | Ans.  a:=3,  or  — 1. 

a;y=:21— 4x?/j •    .  y=l,  or  — 3. 

In  solving  question  18,  let-=v,  and  -=^z\    the   question   then 
X  y 

"becomes  similar  to  the  9th.     In  question  19,  find  the  value  of  xy 

from  the  second  equation,  as  if  it^-ere  a  single  unknown  quantity. 

PROBLEMS  PRODUCIIVG    EQUATIOIVS    OF  THE  SECOIVD   DEGREE, 
CO]\TAIIVL\G   TWO    L^Ki\OVVIV    QUA^fTITIES. 

1.  The  sum  of  two  numbers  is  10,  and  the  sum  of  their  squares 
52  ;  Avhat  are  the  numbers  ?  Ans.  4  and  6. 

2.  The  difference  of  two  numbers  is  3,  and  the  difference  of 
their  squares  39  ;  required  the  numbers.  Ans.  8  and  5. 


EQUATIONS  OF  THE  SECOND  DEGREE.  221 

3.  It  is  required  to  divide  the  number  25  into  two  such  parts, 
that  the  sum  of  their  square  roots  shall  be  7.  Ans.  16  and  9. 

4.  The  product  of  a  certain  number,  consisting  of  two  places, 
by  the  sum  of  its  digits,  is  160,  and  if  it  be  divided  by  4  times 
the  digit  in  unit's  place,  the  quotient  is  4 ;  required  the  number. 

Ans.  32. 

5.  The  difference  between  two  numbers,  multiplied  by  the 
greater,  =16,  but  by  the  less,  =^12 ;  required  the  numbers. 

Ans.  8  and  6. 

6.  Divide  10  into  two  such  parts,  that  their  product  shall  ex- 
ceed their  difference  by  22.  Ans.  6  and  4. 

7.  The  sum  of  two  numbers  is  10,  and  the  sum  of  their  cubes 
is  370;  required  the  numbers.  Ans.  3  and  7. 

8.  The  difference  of  two  numbers  is  2,  and  the  difference  of 
their  cubes  is  98;  required  the  numbers.  Ans.  5  and  3. 

9.  The  sum  of  6  times  the  greater  of  two  numbers,  and  5  times 
the  less,  is  50,  and  their  product  is  20 ;  required  the  numbers. 

Ans.  5  and  4. 

10.  If  a  certain  number,  consisting  of  two  places,  is  divided 
by  the  product  of  its  digits,  the  quotient  will  be  2,  and  if  27  is 
added  to  it,  the  digits  will  be  inverted ;  required  the  number. 

Ans.  36. 

11.  Find  three  such  quantities,  that  the  quotients  arising  from 
dividing  the  products  of  every  two  of  them,  by  the  one  remaining, 
are  a,  6,  and  c.  Ans.  dz]/ab,  zh/ac,  and  ziz^bc. 

12.  The  sum  of  two  numbers  is  9,  and  the  sum  of  their  cubes 
is  21  times  as  great  as  their  sum ;  required  the  numbers. 

Ans.  4  and  5. 

13.  There  are  two  numbers,  the  sum  of  whose  squares  exceeds 
twice  their  product,  by  4,  and  the  difference  of  their  squares  ex- 
ceeds half  their  product,  by  4 ;  required  the  numbers. 

Ans.  6  and  8. 

14.  The  fore  wheel  of  a  carriage  makes  6  revolutions  more  than 
the  hind  wheel,  in  going  120  yards;  but  if  the  circumference  of 
each  wheel  is  increased  1  yard,  it  will  make  only  4  revolutions 
more  than  the  hind  wheel,  in  the  same  distance;  required  the 
circumference  of  each  wheel.  Ans.  4  and  5  yds. 

15.  Two  persons,  A  and  B,  depart  from  the  same  place,  and 
travel  in  the  same  direction ;  A  starts  2  hours  before  B,  and  after 
traveling  30  miles,  B  overtakes  A  ;  but  had  each  of  them  traveled 
half  a  mile  more  per  hour,  B  would  have  traveled  42  miles  before 
overtaking  A.     At  what  rate  did  they  travel  ? 

Ans,  A  2A,  and  B  3  miles  per  hour. 


222  RAY'S   ALGEBRA,    PART   FIRST. 

16.  A  and  B  started  at  the  same  time,  from  two  different  points, 
toward  each  other ;  when  they  met  on  the  road,  it  appeared  that 
A  had  traveled  30  miles  more  than  B.  It  also  appeared,  that  it 
would  take  A  4  days  to  travel  the  road  that  B  had  come,  and  B  9 
days  to  travel  the  road  that  A  had  come.  Find  the  distance  of  A 
from  B,  when  they  set  out.  Ans.  150  miles. 


CHAPTER  VIII. 

PROGRESSIONS  AND  PROPORTION. 

ARITHMETICAL    PROGRESSION. 

Art.  220. — A  seines,  is  a  collection  of  quantities  or  numbers, 
connected  together  by  the  signs  +  or  — ,  and  in  which  any  one 
term  may  be  derived  from  those  which  precede  it,  by  a  rule,  which 
is  called  the  law  of  the  series.     Thus, 

1+3+5+7+9+,  &c., 
2+6+I8+154+,  &c., 
ure  series  ;  in  the  former  of  which,  any  term  may  be  derived  from 
that  which  precedes  it,  by  adding  2;  and  in  the  latter,  any  term 
may  be  found  by  multiplying  the  preceding  term  by  3. 

Art.  221. — An  ArWimetical  Progression  is  a  series  of  quanti- 
ties which  increase  or  decrease,  by  a  common  difference.  Thus, 
the  numbers  1,  3,  5,  7,  9,  &c.,  form  an  increasing  arithmetical 
progression,  in  which  the  common  difference  is  2. 

The  numbers  30,  27, 24,  21 ,  &c.,  form  a  decreasing  arithmetical 
progression,  in  which  the  common  difference  is  3. 

R  E  M  A  R  K. — An  arithmetical  progression  is  termed,  by  some  writers,  an 
equidifferent  series,  or  a, progression  hy  differences. 

Again,  a,  a-\-d,  a-\-2d,  a+3cZ,  a-\-4d,  &c.,  is  an  increasing  arith- 
metical progression,  whose  first  term  is  a,  and  common  difference 
d.  And  if  d  be  negative,  it  becomes  a,  a — d,  a — 2d,  a — Sd,  a — 4<^, 
&c.,  which  is  a  decreasing  arithmetical  progression,  whose  first 
term  is  a,  and  common  difference  d. 

Art.  222.— If  we  take  an  arithmetical  series,  of  which  the 
first  term  is  a,  and  common  difference  d,  we  have 

1st   term  = a 

2d    term  =lst  terra  -\-d=a-{-d 

3d    term  =2d  term  -i-d=a-\-2d 

4th  term  =:3d  term  -{-d=a-\-3d,  and  so  on. 


ARITHxMETICAL   PROGRESSION.  223 

Hence,  the  coefficient  of  d  in  any  term,  is  less  by  unity,  than 
the  number  of  that  term  in  the  series ;    therefore,  the  nth  term 

=a+(n— l)rf. 

If  we  designate  the  ?ith  term  by  Z,  we  have  l=a-^[n—\)d. 
Hence,  the 

RULE, 

FOR  FINDING  ANY  TERM  OF  AN  INCREASING  ARITHMETICAL  SERIES. 

Multiply  the  common  difference  by  the  number  of  tains  less  one, 
and  add  the  product  to  the  first  term ;  the  sum  will  be  the  required 
term. 

If  the  series  is  decreasing,  then  d  is  minus,  and  tl.e  formula  is 
l^=a — {n — \)d.     This  gives  the 

BULK, 

FOR  FINDING  ANY  TERM  OF  A  DECREASING  ARITHMETICAL  SERIES. 

Multiply  the  common  difference  by  the  number  of  terms  less  one, 
and  subtract  the  product  from  the  first  term;  the  remainder  will  be 
the  required  term. 

EXAMPLES. 

1 .  The  first  term  of  an  increasing  arithmetical  series  is  3,  and 
common  difference  5;  required  the  8th  term. 

Here  I,  or  8th  term  =3+ (8—1)5=3 +35=38.     Ans. 

2.  The  first  term  of  a  decreasing  arithmetical  series  is  50,  and 
common  difference  3  ;  required  the  10th  term. 

Here  I,  or  10th  term  =50-(l 0-1)3=50-27=23.     Ans. 

In  the  following  examples,  a  denotes  the  first  term,  and  d  the 
common  difference  of  an  arithmetical  series;  d  being  -plus  when 
the  series  is  increasing,  and  minus  Avhen  it. is  decreasing. 

3.  a=3,  and(7=5;  required  the  6th  term Ans.  28. 

4.  a=20,  and  (?=4;  required  the  15th  term.    .    .    .  Ans.  76. 

5.  a=7,  and  d=\;  required  the  16th  term.  .    .    .    Ans.  10|. 

6.  a=2|-,  and  d=\;  required  the  100th  term.     .    Ans.  35j. 

7.  a=0,  and  d=^;  required  the  11th  term Ans.  5. 

8.  a=30,  and  d=—2;  required  the  8th  term.  .    .    .  Ans.  10. 

9.  a=— 4,  and  <?=3  ;  required  the  5th  term.    .    .    .     Ans.  8. 

10,  a=— 10,  and  cZ=— 2;  required  the  6th  term.     Ans.  —20. 

11.  If  a  body  falls  during  20  seconds,  descending  16j.j  feet  the 
first  second,  48|  feet  the  next,  and  so  on,  how  far  will  it  fall  the 
twentieth  second  ?  Ans.  027^  feet. 

Review. — 220.  What  is  a  series?  Give  examples.  221.  What  is  an 
arithmetical  progression  ?  Give  an  example  of  an  increasing  series.  Of  a 
decreasing  series  ?  222.  What  is  the  rule  for  finding  the  last  term  of  an 
increasing  arithmetical  series  ?  Of  a  decreasing  arithmetical  series  ?  Ex- 
plain the  reason  of  these  rules. 


224  RAY'S   ALGEBRA,    PART   FIRST. 


Art.  233* — Given,  the  first  term  a,  the  common  difierence  d, 
and  the  number  of  terms  n,  to  find  s,  the  sum  of  the  series. 

If  we  take  an  arithmetical  series  of  which  the  first  term  is  3, 
common  difference  2,  and  number  of  terms  5,  it  may  be  written 
in  the  following  forms : 

3,      3+2,    3+4,      3+6,      3+8 
11,    11-2,  11-4,    11-6,    11—8. 
It  is  obvious,  that  the  sum  of  all  the  terms  in  either  of  these 
lines,  will  represent  the  sum  of  the  series ;  that  is, 

s=  3+(  3+2)+(  3+4)+(  3+6)+(  3+8) 
And  g=ll  +  (ll-2)+(ll— 4)  +  (ll-6)+(ll-8) 

Adding,  2^=14+  14        +14        +  14        +  14 
=14X  the  number  of  terms. 
=14X5=70 
Whence,    i'=i  of  70=35. 

Now,  let  Z=  the  last  term,  then  writing  the  series  both  in  a 
direct  and  inverted  order, 

5=a+(«+<^)+(a+2cZ)+(a+3c?)+  .    .    .    .  +1 
And  s^l^{l—d)-{-[l—2d)  +  {l-Zd)-\-,    .    .    .  +a 

By  adding  the  corresponding  terms,  we  have 

25=(;+a)  +  (Z+a)+(Z+a)  +  (/+a.)  .    .  +(Z+a) 

=(J-\-a)  taken  as  many  times  as  there  are  terms  (w)  in 
the  series. 
Hence,     2s-=[l-\-a)n 

This  formula  gives  the  following 

RULE, 

FOR    FINDING    THE    SUM    OF    AN    ARITHMETICAL    SERIES. 

Multiply  half  the  sum  of  the  two  extremes,  by  the  number  of  terms. 

From  the  preceding,  it  appears,  that  the  s^im  of  the  extremes  is 
equal  to  the  sum  of  any  other  two  terms  equally  distant  from  the 
extremes. 

R  E  M  A  R  K. — Since  l=a-\-{n — \)d,  if  we  substitute  this  in  the  place  of  / 

in  the  formula  «=(Z+«)-,  it  becomes  »=  (  2a-\-[n — \)d  J  -.     This  gives 

the  following  Rule,  for  finding  the  sum  of  an  arithmetical  series  :  To  the 
double  of  the  first  term  add  the  product  of  the  nuinber  of  terms  less  one,  by 
the  common  difference,  and  midtiply  the  sum  by  half  the  number  of  terms. 

Reyiew. — 223.  What  is  the  rule  for  finding  the  sum  of  an  arithmeti- 
cal series?     Explain  the   reason  of  the  rule. 


ARITHMETICAL   PROGRESSION.  225 


EXAMPLES. 

1.  Find  the  sum  of  an  arithmetical  series,  of  which  the  first 
term  is  3,  last  term  17,  and  number  of  terms  8. 

^(?^^-)8=80.     Ans. 

2.  Find  the  sum  of  an  arithmetical  series,  whose  first  term  is  1, 
last  term  12,  and  number  of  terms  12.  Ans.  78. 

3.  Find  the  sum  of  an  arithmetical  series,  whose  first  term  is  0, 
common  diflference  1,  and  number  of  terms  20.  Ans.  190. 

4.  Find  the  sum  of  an  arithmetical  series,  whose  first  term  is  3, 
common  difference  2,  and  number  of  terms  21.  Ans.  483. 

5.  Find  the  sum  of  an  arithmetical  series,  whose  first  term  is 
10,  common  difference  — 3,  and  number  of  terms  10.       A.  — 35. 

In  this  case,  the  sum  of  the  negative  terms  exceeds  that  of  the 
positive. 
Art.  224.— The  equations  l=a-{-{n-^l)d  and 

s=^{a-\-l)^,   furnish   the   means  of 

solving  this  general  problem :  Knowing  any  three  of  the  Jive  quan- 
tities a,  d,  n,  I,  s,  which  enter  into  an  arithmetical  series,  to  determine 
the  other  two. 

This  question  furnishes  ten  problems,  the  solution  of  which  pre- 
sents no  difficulty;  for  we  have  always  two  equations,  to  determine 
the  two  unknown  quantities,  and  the  equations  to  be  solved,  are 
either  those  of  the  first  or  second  degree. 

1.  Let  it  be  required  to  find  a  in  terms  of  I,  n,  and  d. 

From  the  first  formula,  by  transposing,  we  have  a=Z — [n — \)d. 
That  is,  the  first  term  of  an  increasing  arithmetical  series  is  equal  to 
the  last  term  diminished  by  the  product  of  the  common  difference  into 
the  number  of  teims  less  one. 

From  the  same  formula,  by  transposing  a,  and  dividing  by  n — 1, 

we  find  d= , . 

w— 1 

That  is,  in  any  arithmetical  seines,  the  co7nmon  difference  is  equal  to 
the  difference  of  the  extremes,  divided  by  the  jiumber  of  terms  less  one. 

Examples,  illustrating  these  principles,  will  be  found  in  the  col- 
lection at  the  close  of  this  subject. 

Review. — 224.  What  are  the  fundamental  equations  of  arithmetical 
progression,  and  to  what  general  problem  do  they  give  rise  ?  To  what  is 
the  first  term  of  an  increasing  arithmetical  series  equal  ?  To  what  is  tho 
common  difference  of  an  arithmetical  series  equal  ? 


226  RAY'S    ALGEBRA,    PART  FIRST. 


Art.  225.— By  means  of  the  preceding  principle,  we  are  ena- 
bled to  solve  the  following  problem. 

Two  numbers,  a  and  b,  being  given,  to  insert  a  number,  m,  of 
arithmetical  means  between  them ;  that  is,  so  that  the  numbers 
inserted,  shall  form,  with  the  two  given  numbers,  an  ai'ithmetical 
series. 

Regarding  a  and  b  as  the  first  and  last  terms  of  an  increasing 
arithmetical  series,  if  we  insert  lyi  terms  between  them,  we  shall 
have  a  series  consisting  of  w-f  2  terms.  But,  by  the  preceding 
principle,  the  common  diflference  of  this  series  will  be  equal  to  the 
difi'erence  of  the  extremes  divided  by  the  number  of  terms  less 

one;  thatis,  £?= — — ; — i-=^-— ,  ;  therefore,  iJie  cojnmon  difference 
m-\-2 — 1     7n-\-l 

will  be  equal  to  the  dijfference  of  the  two  numbers,  divided  by  the  num- 
ber of  means  plus  one. 
Let  it  be  required  to  insert  five  arithmetical  means  between  3 

and  15. 

25 3 

Here  (?=——- =2;  hence  the  series  is  3,  5,  7,  9,  11,  13,  15. 
5+1 

It  is  evident,  that  if  we  insert  the  same  number  of  means  be- 
tween the  consecutive  terms  of  an  arithmetical  series,  the  result 
will  form  a  new  progression.  Thus,  if  we  insert  3  terms  between 
the  consecutive  terms  of  the  progression,  1,  9,  17,  &c.,  the  new 
series  will  be  1,  3,  5,  7,  9,  11,  13,  15,  lY,  and  so  on. 

EXAMPLES. 

1.  Find  the  sum  of  the  natural  series  of  numbers  1,  2,  3,  4,  .  . 
carried  to  1000  terms.  Ans.  500500. 

2.  Required  the  last  term,  and  the  sum  of  the  series  of  odd 
numbers  1,  3,  5,  7,    .    .    .    continued  to  101  terms. 

Ans.  201  and  10201. 

3.  How  many  times  does  a  common  clock  strike,  in  a  week? 

Ans.  1092. 

4.  Find  the  nth.  terra,  and  the  sum  of  n  terms  of  the  natural 
series  of  numbers  1,  2,  3,  4  .    .    .    .  Ans.  ?i,  and  o/i(n+l). 

5.  Find  the  nth.  term,  and  the  sum  of  n  terms,  of  the  series  of 
odd  numbers  1,  3,  5,  7.  Ans.  2?i — 1,  and  li'. 

6.  The  first  and  last  terms  of  an  arithmetical  series  are  2  and 
29,  and  the  common  difierence  is  3;  required  the  number  of  terms 
and  the  sum  of  the  series.  Ans.  10  and  155. 

7.  The  first  and  last  terms  of  a  decreasing  arithmetical  series 
are  10  and  6,  and  the  number  of  terms  9;  required  the  common 
diflFerence,  and  the  sum  of  the  series.  Ans.  J  and  72. 


ARITHMETICAL    PROGRESSION.  227 

8.  The  first  term  of  a  decreasing  arithmetical  series  is  10,  the 
number  of  terms  10,  and  the  sum  of  the  series  85;  required  the 
last  term  and  the  common  difference.  Ans.  7  and  3. 

9.  Required  the  series  obtained  from  inserting  four  arithmetical 
means  between  each  of  the  two  terms  of  the  series  1,  16,  31,  &c. 

Ans.  1,  4,  7,  10,  13,  16,  &c. 

10.  The  sum  of  an  arithmetical  progression  is  72,  the  first  term 
is  24,  and  the  common  difference  is  — 4;  required  the  number  of 
terms.  Ans.  9  or  4. 

In  finding  the  value  of  n  in  this  question,  it  is  required  to  solve 
the  equation  11^ — 13«= — 36,  which  has  two  roots,  9  and  4. 
These  give  rise  to  the  two  following  series,  in  both  of  which  the 
sum  is  72. 

First  series,        24,  20,  16,  12,  8,  4,  0,  —4,  —8. 

Second  series,     24,  20,  16,  12. 

11.  A  man  bought  a  farm,  paying  for  the  first  acre  1  dollar,  for 
the  second  2  dollars,  for  the  third  3  dollars,  and  so  on;  when  ho 
came  to  settle,  he  had  to  pay  12880  dollars  ;  how  many  acres  did 
the  farm  contain,  and  what  was  the  average  price  per  acre? 

Ans.  160  acres,  at  $80|  per  acre, 

12.  If  a  person,  A,  start  from  a  certain  place,  traveling  a  miles 
the  first  day,  2a  the  second,  3a  the  third,  and  so  on ;  and  at  the 
end  of  4  days,  B  start  after  him  from  the  same  place,  traveling 
uniformly  9a  miles  a  day;  when  wuU  B  overtake  A? 

Let  a;=  the  number  of  days  required ;  then  the  distance  traveled 
by  A  in  x  da3'^s  =:aH-2a+3a,  &c.,  to  x  terms,  =Aax(x+l);  and 
the  distance  traveled  by  B  in  [x — 4)  days  =9a(a;— 4). 
Whence  ^ax(a:4-l)=9a(a; — 4).     From  which  a;=8,  or  9. 

Hence,  B  overtakes  A  at  the  end  of  8  days;  and  since,  on  the 
ninth  day,  A  travels  9a  miles,  which  is  B's  uniform  rate,  they  will 
be  together  at  the  end  of  the  ninth  day.  This  is  an  instance  of 
the  precision  with  which  the  solution  of  an  equation  points  out 
the  circumstances  of  a  problem. 

13.  A  sets  out  3  hours  and  20  minutes  before  B,  and  travels  at 
the  rate  of  6  miles  an  hour;  in  how  many  hours  will  B  overtake 
A,  if  he  travel  5  miles  the  first  hour,  6  the  second,  7  the  third,  and 
60  on  ?  Ans.  8  hours. 

14.  Two  travelers,  A  and  B,  set  out  from  the  same  place,  at  the 
same  time.  A  travels  at  the  constant  rate  of  3  miles  an  hour, 
but  B's  rate  of  traveling,  is  4  miles  the  first  hour,  3i  the  second, 
3  the  third,  and  so  on,  in  the  same  series;  in  howmany  hours  will 
A  overtake  B  ?  Ans.  5  hours. 

R  E  V I  E  w. — 22b.  How  do  you  insert  m  arithmetical  means  between  two 
given  numbers  ? 


228  RAY'S   ALGEBRA,    PART    FIRST. 

6KO  METRICAL     PROGRESSION. 

Art.  226. — A  Geometrical  Progression  is  a  series  of  terms, 
each  of  Avhich  is  derived  from  the  preceding,  by  multiplying  it  by 
a  constant  quantity,  termed  the  ratio. 

Thus,  1,  2,  4,  8,  16,  &c.,  is  an  increasing  geometrical  series, 
whose  common  ratio  is  2. 

Also,  54,  18,  6,  2,  &c.,  is  ix  decreasing  geometrical  series,  whose 
common  ratio  is  3. 

Generally,  a,  ar,  ar'^,  ar^,  &c.,  is  a  geometrical  progression,  whose 
common  ratio  is  r,  and  which  is  an  increasing  or  decreasing  series, 
according  as  r  is  greater,  or  less  than  1 . 

It  is  obvious,  that  the  common  ratio  in  any  series,  will  be  ascer- 
tained by  dividing  any  term  of  the  series,  by  that  which  imme- 
diately precedes  it. 

R  E  M  A  R  K. — A  geometrical  progression  is  termed,  by  some  writers,  an 
tquirational  aeries,  or  a  series  of  continued  pro2)ortional8,  or  a  ])rog}-esii{oH 
hy  quotients. 

Art.  227. — To  find  the  last  term  of  the  series. 
Let  a  denote  the  first  term,  r  the  common  ratio,  I  the  nth  term, 
and  5  the  sum  of  n  terms  ;  then,  the  respective  terms  of  the  series 
will  be 

1,    2,     3,     4,      5 n— 3,   n—2,    n—\,        n. 

a,  ar,  at^,  ar^,  ar^ ar^—*,  ar"^-^,  ar"—'^,  ar"—^. 

That  is,  the  exponent  of  r  in  the  second  term  is  1 ,  in  the  third 
term  2,  in  the  fourth  term  3,  and  so  on;  hence,  the  nth  term  of 
the  series  will  be,  l=zai''^—^ ;  that  is, 

Any  term  of  a  geometric  series  is  equal  to  the  product  of  the  first 
term,  hy  the  ratio  raised  to  a  power,  whose  exponent  is  one  less  than 
the  number  of  terms. 

EXAMPLES. 

1.  Find  the  5th  term  of  the  geometric  progression,  whose  first 
term  is  4,  and  common  ratio  3. 

3*=3X3X3X3=81,  and  81X4=324,  the  fifth  term. 

2.  Find  the  6th  term  of  the  progression  2,  8,  32,  &c. 

Aus.  2048. 

3.  Given  the  1st  term  1,  and  ratio  2,  to  find  the  7th  term. 

Ans.  64. 

4.  Given  the  1st  term  4,  and  ratio  3,  to  find  the  10th  term. 

Ans.  78732. 
R  B  V I E  w. — 226.   What  is  a  Geometrical  Progression  ?     Give  examples  of 
an  increasing,  and  of  a  decreasing  geometrical  series.     How  may  the  com- 
mon ratio  in  any  geometrical  series  be  found  ?     227.  How  is  any  term  of  a 
geometrical  series  found  ?     Explain  the  principle  of  this  rule. 


GEOMETRICAL    PROGRESSION.  229 

5.  Find  the  9th  term  of  the  series,  2,  10,  50,  &c.     A.  781250. 

6.  Given  the  1st  term  8,  and  ratio  h,  to  find  the  15th  term. 

Ans.  2Uig- 

7.  A  man  purchased  9  horses,  agreeing  to  pay  for  the  whole 
■what  the  last  would  cost,  at  2  dollars  for  the  first,  6  for  the  second, 
&c. ;  what  was  the  average  price  of  each?  Ans.  $1458. 

Art.  22S. — To  find  the  sum  of  all  the  terms  of  the  series. 

If  we  multiply  any  geometrical  series  by  the  ratio,  the  result 
will  be  a  new  series,  of  which  every  term  except  the  last,  will 
have  a  corresponding  term  in  the  first  series. 

Thus,  let  a,  ar,  ar^,  ai^,  Sec,  be  any  geometrical  series,  and  5  its 

sum,  then     s=a-^ar-{-ar^-{-ai^ +a?"— ^ +«?•"— ^ 

Multiplying  this  equation  by  r,  we  have 

rs=ar-\-a7-'^-{-ar^-\-ar*^ -\-ar"—^-\-ar"^. 

The  terms  of  the  two  series  are  identical,  except  the  Jirst  term 

of  the  first  series,  and  the  last  term  of  the  second  series.     If,  then, 

we  subtract  the  first  equation  from  the  second,  all  the  remaining 

terms  of  the  series  will  disappear,  and  we  shall  have 

rs — s=ai^ — a 

Or  (r— l)s=a(r»— 1) 

„  a(r"-l) 

Hence,  s= 

Since  l=zat''*—^,  we  have         rh 

Therefore,  6'= 

Hence,  the 

RULE, 

FOR    FINDING    THE    SUM    OF    A   GEOMETRICAL    SERIES. 

Multiply  the  last  term  hy  the  ratio,  from  the  product  subtract  th€ 
first  term,  and  divide  the  remainder  by  the  ratiodess  one. 

EXAMPLES. 

1.  Find  the  sum  of  10  terms  of  the  progression  2,  6,  18,  54,  &c. 

The  last  term  =2x3»=:2X  19683-^89366. 

Ir—a    118098—2     -^^.^       , 

.s'= ^= — n — 5 =59048.     Ans. 

/• — 1  6 — 1 

2.  Find  the  sum  of  7  terms  of  the  progression  1,  2,  4,  &c. 

Ans.  127. 

3.  Find  the  sum  of  10  terms  of  the  progression  4,  12,  36,  &c. 

Ans.  118096. 

4.  Find  the  sum  of  9  terms  of  the  series  5,  20,  80,  &c. 

Ans.  436905. 

5.  Find  the  sum  of  8  terms  of  the  series,  whose  first  term  is  6], 
and  ratio  |.  Ans.  307|f  A- 


r-\ 

-ar'' 

ar'^ — a 

rl — a 

r-l 

~r-V 

230  RAY'S    ALGEBRA,    PART   FIRST. 

6.  Find  the  sum  of  8+20+50+,  &c.,  to  7  terms.      A.  3249|. 

7.  Find  the  sum  of  3+4A+t)|+,  &c.,  to  5  terms.        A.  39/(j. 

R  E  M  A  R  K. — If  the  ratio  r  is  less  than  1,  the  progression  is  decreasing, 
and  the  last  term  Ir  is  less  than  a.     In  order  that  both  terms  of  the  fraction 

shall  be  positive,  the  signs  of  the  terms  must  be  changed,  and  we 

r — 1 

have  8= The  sum  of  the  series  when  the  progression  is  decreasing, 

1 — r 

is,  therefore,  found  by  the  same  rule,  as  when  it  is  increasing,  except  that 

the  product  of  the  last  term  by  the  ratio,  is  to  be  subtracted  from  the  first 

term,  and  the  ratio  subtracted  from  unity,  instead  of  subtracting  unity  from 

the  ratio. 

8.  Find  the  sum  of  15  terms  of  the  series  8,  4,  2,  1,  &c. 

Ans.  15ig|J. 

9.  Find  the  sum  of  6  terms  of  the  series  6,  4:1,  3g,  &c. 

Ans.  19i||. 

Art.  229. — The  formula  s=^-^ ;-,  by  separating  the  nume- 
rator into  two  parts,  may  be  placed  under  the  form 

a  ar" 

l—r      1 — r* 
Now,  when  r  is  less  than  1,  it  must  be  a  proper  fraction,  which 

P  I  P  \  ^^     V^ 

may  be  represented  by  -;  then  r'^=  \  -  I    =^.      Since  p  is  less 

than  q,  the  higher  the  power  to  which  the  fraction  is  raised,  the 

less  will  be  the  numerator  compared  with  the  denominator;  that 

is,  the  less  will  be  the  value  of  the  fraction;  therefore,  when  n 

p^ 
becomes  very  large,  the  value  of  — ,  or  ?■**  will  be  veri/  small;  and, 

when  n  becomes  injinitely  great,  the  value  of  -—^,  or  j-",  will  be  in- 

jinitely  small,  that  is,  0.     But,  when  the  numerator  of  a  fraction 

is  zero,  its  value  is  0.     This  reduces  the  value  of  5,"  to  ^j- — .    Hence, 

1 — r 

when  the  number  of  terms  of  a  deci'ea,nng  geometrical  series  is  infi- 

nife,  the  last  term  is  zero,  and  the  sum  is  equal  to  the  frst  term 

divided  hy  one  minus  the  ratio. 

R  E  VIE  w. — 228.  What  is  the  rule  for  finding  the  sum  of  the  terms  of  a 
geometrical  series?  Explain  the  reason  of  this  rule.  When  the  series  is 
decreasing,  how  must  the  formula,  expressing  the  sum,  be  written,  so  that 
both  terms  of  the  fraction  may  be  positive  1  229,  What  is  the  rule  for  find- 
ing the  sum  of  a  decreasing  geometrical  series,  when  the  number  of  terms 
is  infinite?     Explain  the  reason  of  this  rule. 


GEOMETRICAL   PROGRESSION.  231 

1.  Find  the  sum  of  the  infinite  series  1  +  3+44-,  &c. 
Here  a=l,  r=h  and  5=^5 =:r — r=^.     Ans. 

2.  Find  the  sum  of  the  infinite  series  l+2+|+g+,  &c. 

Ans.  2. 

3.  Find  the  sum  of  the  infinite  series' 9+6+4+,  &c.       A.  27. 

4.  Find  the  sum  of  the  infinite  series  1  —  3  +  ?, — 27+'  ^^' 

Ans.  I- 

5.  Find  the  sum  of  the  infinite  series  \-\ — -H — :+  -^+,  &c. 

X^       X*      x^ 

x^ 
Ans.  -T, — i^ . 
x'- — 1 

6.  Find  the  sum  of  the  infinite  geometrical  progression  a — h 
52     ^3     54  .    ,       6  d^ 

A 5-H — 5- — ,  &c.,  in  which  the  ratio  is .  Ans.  — — r. 

a      a^     a*  a  a+o 

7.  If  a  body  moves  10  feet  the  first  second,  5  the  next  second, 
2 2  the  next,  and  so  on,  continually,  how  many  feet  would  it  move 
over  ?  '  Ans.  20. 

Art.  230* — The  two  equations,  l=ar"—^,  and  s= ^-,  fur- 
nish this  general  problem:  knowing  three  of  the  Jive  quantities  a, 
r,  n,  I,  and  s,  of  a  geometrical  progression,  to  determiiie  the  other 
two.  This  problem  embraces  ten  difiierent  questions,  as  in  arith- 
metical progression.  Some  of  the  cases,  however,  involve  the 
extraction  of  high  roots,  the  application  of  logarithms,  and  the 
solution  of  higher  equations  than  have  been  treated  of  in  the  pre- 
ceding pages.  The  following  is  one  of  the  most  simple  and  useful 
of  these  cases. 

Having  given  the  first  and  last  terms,  and  the  number  of  terms 
of  a  geometrical  progression,  to  find  the  ratio. 

Here  Z=ar"-S  or  r"-^— - 

a       

Hence,  '•="-'>/(  ^  )  " 

1.  The  first  and  last  terms  of  a  geometrical  series,  are  3  and 
48,  and  the  number  of  terms  5  ;  required  the  intermediate  terms. 
Here  ?=48,  a=3,  ti— 1=5— 1=4     _ 

Hence,        r*=\f—l(),  and  r'=y'16=4,  and  r=}/4=2. 

2.  In  a.  geometrical  series  of  three  terms,  the  first  and  last 
terms  are  4  and  16;  required  the  middle  term.  Ans.  8. 

In  a  geometrical  progression,  containing  three  terms,  the  middle 
term  is  called  a  mean  proportional  between  the  other  two. 

3.  Find  a  mean  proportional  between  8  and  32.  Ans.  16. 

4.  The  first  and  last  terms  of  a  geometrical  series  are  2  and  1(52, 
and  the  number  of  terms  5;  required  the  ratio.  Ans.  3. 


232  RAY'S   ALGEBRA,    PART   FIRST. 

RATIO    AND    PROPORTION. 

Art.  231. — Two  quantities  of  the  same  kind,  may  be  com- 
pared in  two  ways: 

1st.  By  finding  how  much  the  one  exceeds  the  other. 

2d.   By  finding  how  many  times  the  one  contains  the  other. 

If  we  compare  the  numbers  2  and  6,  by  the  first  method,  we 
say  that  2  is  4  less  than  6,  or  that  6  is  4  greater  than  2. 

If  we  compare  2  and  6,  by  the  second  method,  we  say  that  6  is 
equal  to  three  times  2,  or  that  2  is  one  third  of  6.  This  method  of 
comparison  gives  rise  to  proportion. 

Art.  232* — Ratio  is  the  quotient  which  arises  from  dividing 
one  quantity  by  another  of  the  same  kind.  Thus,  the  ratio  of  2 
to  6  is  3 ;  the  ratio  of  a  to  ma  is  m. 

Remarks. — 1st.  In  comparing  two  numbers  or  quantities  by  their 
quotient,  the  number  expressing  the  ratio  which  the  first  bears  to  the  sec- 
ond, will  depend  on  which  is  made  the  standard  of  comparison.  Thus,  in 
comparing  2  and  6,  if  we  make  2  the  unit  of  measure,  or  standard,  we  find, 
that  6  is  three  times  the  standard.  If  we  make  6  the  unit  of  measure,  or 
standard,  we  find,  that  2  is  one  third  of  the  standard.  In  finding  the  ratio 
of  one  number  to  another,  the  French  mathematicians  always  make  i\iQ  first 
of  the  two  numbers  the  standard  of  comparison ;  while  the  English  make 
the  last  named  the  standard.  Thus,  the  French  say  the  ratio  of  2  to  6  is  3 ; 
while  the  English  say  it  is  J.  The  French  method  is  now  generally  used 
in  the  United  States,  though,  in  a  few  works,  the  other  is  still  retained. 

2d.  In  order  that  two  quantities  may  be  compared,  or  have  a  ratio  to  each 
other,  it  is  evidently  necessary  that  they  should  be  of  the  same  kind,  so  that 
one  may  be  some  part  of,  or  some  number  of  times  the  other.  Thus,  2 
yards  has  a  ratio  to  6  yards,  because  the  latter  is  three  times  the  former ;  but 
2  yards  has  no  ratio  to  6  dollars,  since  the  one  can  not  be  said  to  be  either 
greater,  less,  or  any  number  of  times  the  other. 

Art.  233.— When  tw^o  numbers,  as  2  and  6,  are  compared,  the 
first  is  called  the  antecedent,  and  the  second  the  consequent. 

An  antecedent  and  consequent,  when  spoken  of  as  one,  are 
called  a  couplet.  When  spoken  of  as  two,  they  are  called  the  terms 
of  the  ratio.  Thus,  2  and  6  together,  form  a  couplet,  of  which  2 
is  the  first  term,  and  6  the  second. 

Art.  234. — Ratio  is  expressed  in  two  ways. 

Ist.  In  the  form  of  a  fraction,  of  which  the  antecedent  is  the 
denominator,  and  the  consequent  the  numerator.  Thus,  the  ratio  of 
2  to  6,  is  expressed  by  4;  the  ratio  of  3  to  12,  by  ^^,  &c. 

Review, — 231.  In  how  many  ways,  may  two  quantities  of  the  same 
kind  be  compared?  Compare  the  numbers  2  and  6  by  the  first  method. 
By  the  second.  232.  What  is  ratio  ?  Give  an  illustration.  233.  When 
two  numbers  are  compared,  what  is  the  first  called?  The  second?  Q-ive 
an  example. 


RATIO   AND   PROPORTION.  233 


2d.  By  placing  two  points  (:)  between  the  terms  of  the  ratio. 
Thus,  the  ratio  of  2  to  6,  is  written  2:6;  the  ratio  of  3  to  8, 
3  :  8,  &c. 

Art.  235.-^The  ratio  of  two  quantities,  may  be  either  a  whole 
number,  a  common  fraction,  or  an  interminaie  decimal. 

Thus,  the  ratio  of  2  to  6  is  f ,  or  3. 


The  ratio  of  10  to  4  is 


TO. 


The  ratio  of  2  to  |/5  is  ^-,  or  ?:?|5±,  or  1.1 18+. 

We  see,  from  this,  that  the  ratio  of  two  quantities  can  not 
always  be  expressed  exactly,  except  by  symbols  ;  but,  by  taking  a 
sufficient  number  of  decimal  places,  it  may  be  found  to  any  re- 
quired degree  of  exactness. 

Art.  236. — Since  the  ratio  of  two  numbers  is  expressed  by  a 
fraction,  of  which  the  antecedent  is  the  denominator,  and  the  con- 
sequent the  numerator,  it  follows,  that  whatever  is  true  with  regard 
to  a  fraction,  is  true  with  regard  to  the  terms  of  a  ratio.     Hence, 

1st.  To  multijjly  the  consequent,  or  to  divide  the  antecedent  of  a 
ratio  hy  any  number^  multiplies  the  ratio  by  that  number.  (Articles 
122,  125.) 

Thus,  the  ratio  of  4  to  12,  is  3. 

The  ratio  of  4  to  12X5,  is  3X5. 

The  ratio  of  4-T-2  to  12,  is  6,  which  is  equal  to  3X2. 

2d.  To  divide  the  consequent,  or  to  midtiply  the  antecedent  of  a  ratio 
by  any  number,  divides  the  ratio  by  that  number.  (Articles  123, 
124.) 

Thus,  the  ratio  of  3  to  24,  is  8. 

The  ratio  of  3  to  24h-2,  is  4,  which  is  equal  to  8-T-2. 

The  ratio  of  3X2  to  24,  is  4,  which  is  equal  to  8-^2. 

3d.  To  multiply,  or  divide,  both  the  antecedent  and  consequent  of 
a  ratio  by  any  number,  does  not  alter  the  ratio.     (Articles  126, 127.) 

Thus,  the  ratio  of  6  to  18,  is  3. 

The  ratio  of  6X2  to  18X2,  is  3. 

The  ratio  of  6-r-2  to  18-^2,  is  3. 

Art.  23'7. — When  the  two  numbers  are  equal,  the  ratio  is  said 
to  be  a  ratio  of  equality.     When  the  second  number  is  greater  than 

Review. — 234.  When  are  the  antecedent  and  consequent  of  a  ratio 
called  a  couplet?  When  the  terras  of  a  ratio?  By  what  two  methods  is 
ratio  expressed  ?  Give  an  example.  235.  What  forms  may  the  ratio  of  two 
quantities  have  ?  236.  How  is  a  ratio  affected  by  multiplying  the  conse- 
quent, or  dividing  the  antecedent?  Why?  Howls  a  ratio  affected  by 
dividing  the  consequent,  or  multiplying  the  antecedent?  Why?  How  is 
a  ratio  affected,  by  either  multiplying  or  dividing  both  antecedent  and 
consequent  by  the  same  number  ?  Why  ? 
20 


234  RAY'S  ALGEBRA,   PART  FIRST. 

the  first,  the  ratio  is  said  to  be  a  ratio  of  gr^eater-  inequality,  and 
when  it  is  less,  the  ratio  is  said  to  be  a  ratio  of  less  inequality. 

Thus,  the  ratio  of  4  to  4,  is  a  ratio  of  equality. 

The  ratio  of  4  to  8,  is  a  ratio  of  greater  inequality. 

The  ratio  of  4  to  2,  is  a  ratio  of  less  inequality. 

We  see,  from  this,  that  a  ratio  of  equality  may  be  expressed 
by  1;  a  ratio  of  greater  inequality,  by  a  number  greater  than  1; 
and  a  ratio  of  less  inequality,  by  a  number  less  than  1. 

Art.  SJS8. — When  the  corresponding  terms  of  two  or  more 
ratios  are  multiplied  together,  the  ratios  are  said  to  be  compounded, 
and  the  result  is  termed  a  compound  ratio.  Thus,  the  ratio  ^3**, 
compounded  with  the  ratio  f ,  is  ^f  X5=f  §=4.  In  this  case,  3 
multiplied  by  5,  is  said  to  have  to  10  multiplied  by  6,  the  ratio 
compounded  of  the  ratios  of  3  to  10  and  5  to  6. 

Art.  239. — Ratios  may  be  compared  with  each  other,  by  re- 
ducing the  fractions  which  represent  them,  to  a  common  denom- 
inator. Thus,  to  ascertain  whether  the  ratio  of  2  to  5  is  greater 
than  the  ratio  of  3  to  8,  we  have  the  two  fractions,  #  and  |,  which 
being  reduced  to  a  common  denominator,  are  ^g'  and  ^g* ;  and, 
since  the  first  is  less  than  the  second,  we  infer,  that  the  ratio  of  2 
to  5  is  less  than  the  ratio  of  3  to  8. 

PROPORTION. 

Art.  240. — Proportion  is  an  equality  of  ratios.     Thus,  if  a,  b,  c, 

d  are  four  quantities,  such  that  -  is  equal  to  -,  then  a,  b,  c,  d  form 

a  proportion,  and  we  say  that  a  is  to  b,  as  c  is  to  cZ ;  or,  that  a  has 
the  same  ratio  to  b,  that  c  has  to  d. 

Proportion  is  written  in  two  ways. 

1st.  By  placing  the  double  colon  between  the  ratios.     Thus, 
a  :  b  '.  :  c  :  d. 

2d.  By  placing  the  sign  of  equality  between  them.     Thus, 
a  :  b^^c  :  d. 

The  first  method  is  the  one  generally  used. 

From  the  preceding  definition,  it  follows,  that  when  four  quan- 
tities are  in  proportion,  the  second  divided  by  the  first,  gives  the 
same  quotient  as  the  fourth  divided  by  the  third.  This  is  the  test 
of  the  proportionality  of  four  quantities.     Thus,  if  3,  6,  5,  10  are 

Re  VIE  TV. — 237.  What  is  a  ratio  of  equality?  Of  greater  inequality? 
Of  less  inequality  ?  Give  examples.  238.  When  are  two  or  more  ratios 
said  to  be  compounded  ?  Give  an  example.  239.  How  may  ratios  be  com- 
pared to  each  other  ?  240.  What  is  proportion  ?  Give  an  example.  Hovr 
are  four  quantities  in  proportion  written  ?     Give  examples. 


RATIO  AND   PROPORTION.  235 

the  four  terms  of  a  true  proportion,  so  that  3  :  6  :  :  5  :  10,  we 
must  have  |=  5^. 

If  these  fractions  are  equal  to  each  other,  the  proportion  is  true; 
if  they  are  not  equal  to  each  other,  it  is  false. 

Thus,  let  it  be  required  to  find  whether  3  :  8  :  :  2  :  5. 

The  first  ratio  is  |,  the  second  is  |,  or  g^,  and  ^^ ;  therefore, 
3,  8,  2,  5  are  not  proportional  quantities. 

R  E  M  A  R  K. — The  words  ratio  and  proportion,  in  common  language,  aro 
sometimes  confounded  with  each  other.  Thus,  two  quiffitities  aro  said  to  be 
in  the  proportion  of  3  to  4,  instead  of,  in  the  ratio  of  3  to  4.  A  ratio  sub- 
sists between  two  quantities,  a  proportion  only  between  /our.  It  requires 
two  equal  ratios  to  form  a  proportion. 

Art.  241. — In  the  proportion  a  :  b  :  :  c  :  d,  each  of  the  quan- 
tities a,  b,  c,  d,  is  called  a  iei-m.  The  first  and  last  terms  are 
called  the  extremes,  the  second  and  third,  the  means. 

Art.  242. — Of  four  proportional  quantities,  the  first  and  third 
are  called  antecedents,  and  the  second  and  fourth,  consequents  (Art. 
233);  and  the  last  is  said  to  be  a  fourth  proportional  to  the  other 
three,  taken  in  their  order. 

Art.  243. — Three  quantities  are  in  proportion,  when  the  first 
has  the  same  ratio  to  the  second,  that  the  second  has  to  the  third. 
In  this  case,  the  middle  term  is  called  ^  mean  proportional  hQtv^Qen 
the  other  two.     Thus,  if  we  have  the  proportion 

a:b  :  '.  b  '.  c 
then  b  is  called  a  mean  proportional  between  a  and  c,  and  c  is  called 
a  third  proportional  to  a  and  b. 

Art.  244,— Proposition  I. — In  every  proportion,  the  product  of 
the  means  is  equal  to  the  product  of  the  extremes. 

Let  a  :  b  :  :  c  :  d. 

Then,  since  this  is  a  true  proportion,  the  quotient  of  the  second 
divided  by  the  first,  is  equal  to  the  quotient  of  the  fourth  divided 
by  the  third.     Therefore,  we  must  have 
b^d 
a     c 
Multiplying  both  sides  of  this  equality  by  ac,  to  clear  it  of  frac- 
abc     adc       „     ,  . 

tions,  we  have  — = .     Or,  bc=^ad. 

a        c 

Illustration  by  numbers.     3  :  G  :  :  5  :  10,  and  6X5=3X10. 

R  E  V I E  w. — 240.  Give  examples  of  a  true  and  false  proportion.  What 
is  a  test  of  the  proportionality  of  four  quantities  ?  241.  What  are  the  first 
and  last  terms  of  a  proportion  called  ?  The  second  and  third  terms  ? 
242.  What  are  the  first  and  third  terms  of  a  proportion  called  ?  The  sec- 
ond and  fourth?  243.  When  are  three  quantities  in  proportion  ?  Give  an 
example.     What  is  the  second  term  called  ?     The  third  ? 


236  RAY'S   ALGEBRA,    PART    FIRST. 

From  the  equation  bc=ad,  we  have  d= — ,  c=—,  6= — ,  and  a=-3> 

Q/  0  C  0/ 

from  which  we  see,  that  if  any  three  terms  of  a  proportion  are 
given,  the  fourth  may  be  readily  found. 

The  first  three  terms  of  a  proportion,  are  ac,  hd,  and  acxy ;  what 
is  the  fourth?  Ans.  bdxy. 

Remark  . — This  proposition  furnishes  a  more  convenient  test  of  the 
proportionality  of  four  quantities,  than  the  method  given  in  Article  240. 
Thus,  to  ascertain  wliether  3  :  8  :  :  2  :  5,  it  is  merely  necessary  to  compare 
the  product  of  the  means  and  the  extremes;  and,  since  3X5  is  not  equal 
to  8X2,  we  infer  that  the  proportion  is  false. 

Art.  245. — Proposition  II. — Conversely,  If  the  product  of  two 
quantities  is  equal  to  the  product  of  two  others,  two  of  them  may  he 
made  the  means,  and  the  other  two  the  extremes  of  a  proportion. 

Let  hc^=ad. 

Dividing  each  of  these  equals  by  ac,  we  have 

he  ad  ^  h  d 
—=—;  Or,  -=-. 
ac     ac  a     c 

That  is,  a-.h-.'.c  d. 

Illustration.     5X8=4X10,  and  4  :  5  :  :  8  :  10. 

Art.  246. — Proposition  III. — If  three  quantities  are  in  contin- 
ued proportion,  the  product  of  the  extremes  is  equal  to  the  sq}iare  of 
the  mean. 

If  a  :  5  :  :  6  :  c 

Then,  by  Art.  244,  ac=bh=.b'\ 

It  follows,  from  Art.  245,  that  the  converse  of  this  proposition 
is  also  true.     Thus,  if  ac=¥, 

Then,  a  :  b  :  :  b  :  c. 

That  is,  if  the  product  of  the  first  and  third  of  two  quantities,  is 
equal  to  the  square  of  a  second,  thefrstis  to  the  second,  as  the  second 
is  to  the  third. 

Illustration.     IT  4  :  6  :  :  6  :  9,  then  4X9=6^=36. 
If  2X8=16,  then  2  :  ^16  :  :  ^/m  :  8 

Or  2  :  4  :  :  4  :  8. 

Art.  247.— Proposition  IV. — If  four  quantities  are  inpropor- 
lion,  they  will  be  in  proportion  by  ALTEni<!ATiois! ;  that  is,  the  first  will 
have  the  same  ratio  to  the  third,  that  the  secmid  has  to  the  fourth. 

Let  a  '.  b  :  :  c  :  d. 

m.  .      •  ^    d 

lhi8  gives,  -=:-. 

°  a     c 

be 
Multiplying  both  sides  by  c,        —=d. 


RATIO   AND   PROPORTION.  237 

c    d 
Dividing  both  sides  by  6,        -=?-. 

That  is,  a-,  c  :  '.  h  :  d. 

lUustration.     2  :  7  :  :  6  :  21,  and  2  :  6  :  :  7  :  21. 

Art.  248. — Proposition  V. — If  four  quantities  are  in  propor- 
tion, they  will  he  in  proportion  hy  inversion  ;  that  is,  the  second  will 
be  to  the  first  as  the  fourth  to  the  third. 

Let  a  :  b  :  :  c  :  d. 

By  Art.  244,  ad=bc. 

Dividing  both  sides  by  b,      'ir=^' 

Dividing  both  sides  by  d,        jr^^- 

That  is,  b  :  a  :  :  d  :  c. 

Illustration.     2  :  5  :  :  6  :  15,  and  5  :  2  :  :  15  :  6. 

Art.  249. — Proposition  YI. — If  two  sets  of  proportions  have 
an  antecedent  and  consequent  in  the  one,  equal  to  an  antecedent  and 
consequent  in  the  other,  the  remaining  terms  will  be  proportional. 


Let 

a'.b'.'.c-.d    (1.) 

And 

a-.b-.'.e-.f    (2.) 

Then  will 

c:d'.'.e:f 

For,  from  1st  proportion 

b    d 
a~~V 

From  2d  proportion, 

a     e' 

Hence, 

c     e' 

This  gives, 
Illustration. 

c:d::e:f 
3:5::  6:  10 

3 

:     5  :  :  9 :  15 

And 

6 

:  10  :  :  9  :  15. 

Remark  . — This  proposition  is  generally  termed  equality  of  ratios.  It 
is  almost  self-evident. 

Art.  250. — Proposition  VII. — If  four  quantities  are  in  propor- 
tion, they  ivill  he  in  proportion  hy  composition;  that  is,  the  sum  of 
the  first  and  second,  loill  be  to  the  second,  as  the  sum  of  the  third  and 
fourth,  is  to  the  fourth. 

Let  a  :  h  :  :  c  :  d 

Then  will  a+b  :  b  :  :  c-\-d  :  d 

From  the  1st  proportion,       be— ad,  by  Art.  244. 


238  RAY'S   ALGEBRA,    PART   FIRST. 

Adding  hd  to  each,  bd=bd, 

bc-\-bd=ad+bd',     Or  b{c+d)=d{a+b). 

Dividing  each  side  by  c-{-d,         b=  —7-7^ ; 
Bya+6.  *  '^ 


a-\-b     c+d* 
This  gives,  a-\-b  :  b  :  :  c-{-d  :  d. 

Illustration.  3  :  4  :  :  6  :  8 

3+4:4::  6+8:8;     Or,  7  :  4  :  :  14  :  8. 

Remark  . — In  a  similar  manner,  it  may  be  proved,  that  the  sum  of  tho 
first  and  second  terms,  will  be  to  the  first,  as  the  sum  of  the  third  and 
fourth  is  to  the  third. 

Art.  251. — Proposition  VIII. — If  four  quantities  are  in  pro- 
portion, they  will  be  in  proportion  bij  division  ;  that  is,  the  difference 
of  the  first  and  second,  will  be  to  the  second,  as  the  difference  of  the 
third  and  fourth  is  to  the  fourth. 

Let  a  :  b  :  :  c  :  d. 

Then  will  a — b  :  b  :  :  c — d  :  d. 

From  the  1st  proportion,       bc^=^ad,  by  Art.  244. 
Subtracting  bd  from  each,    bd=bd 

be — bd=ad — bd ; 
Or,  b[c—d)=d[a—b). 

Dividing  each  side  by  c — d,    b=— — -7-  ; 

Ti  J.  b  d 

By  a—b  r= ,. 

''  a—b     c—d 

This  gives,  a — b  :  b  :  i  c — d  :  d. 

Illustration.  8:5::  16  :  10 

8-5:  5:  :  16-10:  10;     Or,  3  :  5  :  :  6  :  10. 

Remark  . — In  a  similar  manner,  it  may  be  proved,  that  the  diflferenco 
of  the  first  and  second  will  be  to  the^r»^,  as  the  difference  of  the  third  and 
fourth  is  to  the  third. 

Art.  252. — Proposition  IX. — If  four  quantities  are  in  propor- 
tion, the  sum  of  the  first  and  second  will  be  to  their  difference,  as  the 
sum  of  the  third  and  fourth  is  to  their  difference. 

Let  a  :  b  :  :  c  :  d,     (1.) 

Then  will  a+6  :  a — b  :  :  c-{-d  :  c — d. 

From  the  1st,  by  composition.  Art.  250, 

a+6  :  b:  :  c-{-d  :  d. 
By  alternation,  a+6  :  c-\-d  :  :  b  :  d,  Art.  247. 


RATIO   AND   PROPORTION.  239 


This  gives, 

c:\-d_d 
a+6     b' 

From  the  1st,  by  division, 

a—b  :  b  :  :  c—d  :  d, 

By  alternation. 

a — b  :  c — d  :  :  b  :  d; 

This  gives, 

c—d    d    .           c-\-d     c—d 

r=r ;  hence  — —= r. 

a — b    b              a+6    a — 6 

That  is, 

a+6  :  c+(Z  :  :  a — 6  :  c — d; 

Or,  by  alternation, 

a+6  :  a— 6  :  :  c-{-d  :  c—d. 

Illustration. 

5:3::  10:  6 

5+3  :  5-3  :  :  10+6  :  10-6 
Or,  8:2::  16  :  4. 

Art.  253. — Proposition  X. — If  four  quantities  at-e  in  proporiioriy 
like  powerSy  or  roots,  of  those  quantities,  will  also  be  in  proportion. 

Let  a  :  b  :  :  c  :  d. 

Then  v^ill  a"  :  6"  :  :  c"  :  c^« 

For,  since  -=-. 

a     c 

If  we  raise  each  of  these  equals  to  the  nth  power,  we  have, 

6"  _d'* 

a"  "~c"  * 

That  is,  a"  :  6«  :  :  c«  :  d", 

AVhere  n  may  either  be  a  whole  number  or  a  fraction. 

Illustration.  2  :  3  :  :  4  :  6 

2^  :  32  :  :  4^  :  6^ 

Or  4  :  9  :  :  16  :  36 

Also,  a^  :  6^  :  :  m^d'  :  mW 

And  l/a*  :  V'b'  :  :  >/ w^'  :  V^^^ 

Or  a  :  b  :  :  ma  :  mb. 

Art.  254. — Proposition  XI. — If  two  sets  of  quantities  are  in 
iwoportion,  tlie  products  of  the  corresponding  terms  will  also  be  in 
•proportion. 

Let  a  :  b  :  :  c  :  d,     (1.) 

And  m  :  n  :  :  r  :  s;     (2.) 

Then  will  am  :  bn  :  :  cr  :  ds. 

For,  from  the  1st,       -=- ;  and  from  the  2d,     — =-. 
'  a     c  m     r 

Multiplying  these  equals  together, 

6,  ,n  d^s  bn  ds 
-X — =-X-,  or  — = — . 
am     c     r       am     cr 

This  gives,  am  \hn\:  cr  :  ds. 


240  RAY'S   ALGEBRA,    PART   FIRST. 

Illustration. 


3:5: 

:  6  :  10, 

4:3: 

:  8  :  6, 

2:  15: 

:  48  :  60. 

Dividing  by  a-{-c-\-m, 


Art.  255*  —  Proposition  XII. —  In  any  continued  proportion, 
that  is,  any  number  of  proporiions  having  the  same  ratio,  any  one 
antecedent  is  to  its  consequent,  as  the  sum  of  all  the  antecedents  is 
to  the  sum  of  all  the  consequents. 

Let  a  :  h  :  :  c  '.  d  '.  :  in  \  n,  &c. 

Then  will  a  :  b  :  :  a-\-c-\-m  :  6+d+n ; 

Since  a  :  b  :  :  c  :  d,  we  have      bc^:^ad. 
Since  a  :  b  :  :  m  :  n,  we  have  bm^=an. 

Adding  ab  to  each,  ab^^ab.     The  sums  of  these  equali- 

ties give  ab-\'bc-\-bm~ab-\-ad-\-an ; 

Or  b[a+c-\-m)=a{b+d-\-n). 

J a{b-{-d-\-n)  ^ 

Dividing  both  sides  by  a  -=^— . 

^  •'a    a-i-c-\-m 

This  gives,  a  :  b  :  :  a-\-c^m  :  b-\-d-{-7i. 

Illustration.  3  :  4  :  :  (3  :  8  :  :  9  :  12 

3:4::  3+6+9  :  4+8+12 
Or  3  :  4  :  :  18  :  24. 

Remark  . — In  the  preceding  demonstrations,  the  proof  has  generally 
been  made  to  involve  the  definition  of  proportion,  that  is,  that  the  four 

b      d 
quantities,  a,  b,  c,  d,  are  in  proportion,  Avhen  -=-.     This  is  regarded  aa 

a  matter  of  great  importance  to  the  pupil.  If  the  instructor  chooses  to  dis- 
pense with  this,  as  some  writers  do,  several  of  the  demonstrations  may  bo 
somewhat  shortened.  There  arc  several  other  Propositions  in  Proportion, 
that  may  be  easily  demonstrated,  in  a  manner  similar  to  the  preceding,  but 
they  are  of  so  little  use,  as  not  to  be  worthy  of  the  pupil's  attention. 


NOW    PUBLISHED. 

RAY'S  ALGEBRA,   PART  II.-HIGIIER  ALGEBRA. 

RAY'S  ALGEBRA,  PART  SECOND,  for  advanced  students,  contains  a  concise  re- 
Tiew  of  the  elementary  principles  presented  in  part  first,  with  more  dilTicult  exam- 
ples for  practice.  Also,  a  full  discussion  of  the  higher  practical  parts  of  the  science, 
embracing  the  General  Theory  of  equations,  with  Sturm's  celebrated  theorem  illus- 
trated by  examples ;  Horxkr's  method  of  resolving  numerical  equations,  &c.,  &c. 
Designed  to  be  a  thorough  treatise  for  High  Schools  and  for  Colleges.  The  author  has 
endeavored  to  present  every  subject  in  a  plain  and  simple  manner,  with  numerous 
interesting  and  appropriate  illustrations  and  examples. 

THE     END. 


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